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Research Article Evaluation of Density Matrix and Helmholtz Free Energy...
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Research ArticleEvaluation of Density Matrix and Helmholtz FreeEnergy for Harmonic Oscillator Asymmetric Potential viaFeynmans Approach
Piyarut Moonsri1 and Artit Hutem2
1 Chemistry Division Faculty of Science and Technology Phetchabun Rajabhat University Phetchabun 67000 Thailand2 Physics Division Faculty of Science and Technology Phetchabun Rajabhat University Phetchabun 67000 Thailand
Correspondence should be addressed to Artit Hutem magoohootemyahoocom
Received 18 August 2014 Revised 8 October 2014 Accepted 9 October 2014 Published 2 November 2014
Academic Editor Miquel Sola
Copyright copy 2014 P Moonsri and A Hutem This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We apply a Feynmans technique for calculation of a canonical density matrix of a single particle under harmonic oscillatorasymmetric potential and solving the Bloch equation of the statistical mechanics system The density matrix (P
119906) and kinetic
energy per unit length (120591119871) can be directly evaluated from the solving solutions From the evaluation it was found that both of the
density matrix and kinetic energy per unit length depended on the parameter of the value of asymmetric potential (120582) the value ofaxes-shift potential (119892) and temperature (T) Comparison of the Helmholtz free energy was derived by the Feynmans techniqueand the path-integral method The results illustrated are slightly different
1 Introduction
We consider a quantum mechanical ensemble of identicalsystems in situations where the description is incompleteWhen the description of the system is incomplete the state ofthe system is described by means of what is called a densitymatrix In the statistical point of view a state of quantumensemble is described by a density operator (or statisticaloperator) P
119906 the corresponding matrix is called a density
matrix A knowledge of this matrix enables us to calculatethe mean value of physical quantity of the system and alsothe probabilities of various values of such a quantity
March and Murray used the Bloch equation to calculatethe canonical density matrix in a single particle frameworkwhich may be related directly to the generalized canonicaldensity matrix containing the Fermi-Dirac function [1]Howard et al used the Bloch equation evaluate the electrondensity P
119906and kinetic energy 120591
119871(119909) for the one dimensional
sech2(119909) potential Howard et al used tool employed is theSlater sum which satisfies a partial differential equation [2]
For calculation of the canonical density matrix in termsof the Slater sum and kinetic energy per unit length in terms
of the Slater sum see March and Nieto [3] To study problemsof the canonical ensemble we focus on the calculation thecanonical density matrix and the Helmholtz free energyof a single particle under asymmetric harmonic oscillatorpotential
The scheme of the paper is as follows In Section 2 thereis detailing with the density matrix In Section 3 we show thecalculation of the canonical density matrix (or charge bond-order matrix) and the Helmholtz free energy by Feynmansapproach In Section 4 we representation of the evaluationof the density matrix and the Helmholtz free energy bythe path-integral method (120581(119910)) method In Section 5 wemake a presentation of numerical results for the Helmholtzfree energy of a single particle under asymmetric harmonicoscillator potential The conclusion and discussion are givenin Section 6
2 Statistical Density Matrix
A density matrix is a matrix that describes a quantumsystem in a mixed state a statistical ensemble of severalquantum states This should be contrasted with a single state
Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2014 Article ID 912054 9 pageshttpdxdoiorg1011552014912054
2 Advances in Physical Chemistry
vector that describes a quantum system in a pure state Thedensity matrix is the quantum-mechanical analogue to aphase-space probability measure (probability distribution ofposition and momentum) in classical statistical mechanicsExplicitly suppose a quantum system may be found in state10038161003816100381610038161205951⟩ with probability 119875
1or it may be found in state |120595
2⟩ with
probability1198752or it may be found in state |120595
3⟩with probability
1198753 and so on The density matrix for this system is
P = sum119899
119875119899
1003816100381610038161003816120595119899⟩ ⟨1205951198991003816100381610038161003816 (1)
where (|120595119899⟩) need not be orthogonal and sum
119899119875119899= 1 By
choosing an orthonormal basis (|119906119895⟩) one may resolve the
density operator into the density matrix whose elements are
P119894119895= sum119899
119875119899⟨119906119895| 120595119899⟩ ⟨120595
119899| 119906119894⟩ (2)
For an operator A (which describes an observable A of thesystem) the expectation value ⟨A⟩ is given by
⟨A⟩ = sum119899
119875119899⟨120595119899
10038161003816100381610038161003816A10038161003816100381610038161003816120595119899⟩ = sum
119894
⟨119906119894
10038161003816100381610038161003816PA
10038161003816100381610038161003816119906119894⟩ = tr (PA)
(3)
In the canonical ensemble the probability that the system isin state |119899⟩ is given by
119875119899=
119890minus120573E119899
sum119899119890120573E119899
(4)
and the thermal average of an arbitrary operator A is
⟨A⟩ = sum119899
⟨11989910038161003816100381610038161003816A10038161003816100381610038161003816119899⟩(
119890minus120573E119899
sum119899119890120573E119899
)
=1
Q119873(120573)
sum119899
⟨11989910038161003816100381610038161003816A10038161003816100381610038161003816119899⟩ 119890
minus120573E119899
(5)
The operator
P =1
Q119873(120573)
sum119899
|119899⟩ ⟨119899| 119890minus120573E119899 (6)
is commonly referred to as density operator and its matrixrepresentation as the density matrix Here Q
119873(120573) is the
partition function
3 A Linear Harmonic Oscillator AsymmetricPotential by Feynmans Approach
Next we consider the case of a linear harmonics oscillator ofasymmetric potential whose Hamiltonian is given by [4]
H119878=2
2119898+1
21198981205962
1199092
+ radic119898120596
ℎ
ℎ120596
2120582(119909 minus 119892radic
ℎ
119898120596) (7)
we note that this is a time-independent Hamiltonian ofharmonic oscillator asymmetric axes-shift potential system
Substituting in the time-independent Hamiltonian equation(7) into the Bloch equation [3] we can rewrite the differentialequation completely in terms of P
119906(120585 120585 119891) as
minus120597P119906
120597120573= minus
ℎ2
2119898
1205972P119906
1205971199092+1
21198981205962
1199092
P119906
+ radic119898120596
ℎ
ℎ120596
2120582(119909 minus 119892radic
ℎ
119898120596) P
119906
(8)
We define some new dimensionless variable The positionvariable 119909 = 119909 is replaced with the dimensionless variable120585
120585 = radic119898120596
ℎ119909 119909
2
=ℎ
1198981205961205852
120573 =2119891
ℎ120596 (9)
Substituting in for 119909 in terms of 120585 we can rewrite thedifferential equation completely in terms of 120585 and P
119906as
minus120597P119906
120597119891= minus
1205972P119906
1205971205852+ 1205852
P119906+ 120582120585P
119906minus 120582119892P
119906 (10)
Knowing the Gaussian dependence of the density matrix P119906
in case of the free particle we try the following form
P119906(120585 120585 119891) asymp 119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
120582119892P119906= 120582119892119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
minus120597P119906
120597119891= [120585
2119889119886 (119891)
119889119891+ 120585
119889119887 (119891)
119889119891+119889119888 (119891)
119889119891]
times 119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
minus1205972P
119906
1205971205852= (2119886 (119891) minus 4119886
2
(119891) 1205852
minus 4119886 (119891) 119887 (119891) 120585 minus 1198872
(119891))
times 119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
1205852
P119906= 1205852
119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
120582120585P119906= 120582120585119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
(11)
Substituting expansion equation (11) into (10) leads us to theequation
119889119886 (119891)
1198891198911205852
+119889119887 (119891)
119889119891120585 +
119889119888 (119891)
119889119891
= (1 minus 41198862
(119891)) 1205852
+ (120582 minus 4119886 (119891) 119887 (119891)) 120585
+ (2119886 (119891) minus 120582119892 minus 1198872
(119891))
(12)
Advances in Physical Chemistry 3
Equating the coefficients for 1205852 120585 and 1205850 = 1 on both sides ofthe last equation (12) gives
119889119886 (119891)
119889119891= (1 minus 4119886
2
(119891)) (13)
119889119887 (119891)
119889119891= (120582 minus 4119886 (119891) 119887 (119891)) (14)
119889119888 (119891)
119889119891= (2119886 (119891) minus 120582119892 minus 119887
2
(119891)) (15)
respectively Computing (13) is integrated to give
119886 (119891) =1
2coth (2119891) (16)
Equation (14) is the linear first-order ordinary differentialequation to give [5 6]
120582 =119889119887 (119891)
119889119891+ 2 coth (2119891) 119887 (119891)
119887 (119891) = 119890minusint 2 coth(2119891)119889119891
(int120582119890int 2 coth(2119891)119889119891
119889119891 + 119860)
119887 (119891) =120582
2coth (2119891) + 119860csch (2119891)
(17)
Equation (15) is integrated with respect to 119891 as119889119888 (119891)
119889119891= (coth (2119891) minus 120582119892 minus (120582
2coth (2119891) +119860csch (2119891))
2
)
119888 (119891) = int coth (2119891) 119889119891 minus int120582119892119889119891
minus int(120582
2coth (2119891) + 119860csch (2119891))
2
119889119891
119888 (119891) =1
8(minus2119891120582 (4119892 + 120582) + (4119860
2
+ 1205822
) coth (2119891)
+ 4 ln (sinh (2119891))
+2119860120582csch (119891) sech (119891) minus ln119861) (18)
where 119861 120582 and 119892 are constants Substituting 119886(119891) 119887(119891) and119888(119891) of (16) (17) and (18) respectively into (11) we mayrewrite density matrix asP119906
=119861
radicsinh (2119891)
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
)) coth (2119891)
+(120585119860 +119860120582
2) csch (2119891) minus
119891120582
4(4119892 + 120582) ])
(19)
Thus the Taylor series expansion of sinh(2119891) coth(2119891) about119891 rarr 0 is given by
lim119891rarr0
sinh (2119891) asymp 2119891 lim119891rarr0
coth (2119891) asymp 1
2119891 (20)
we obtain the density matrix final form
P119906asymp
119861
radic2119891
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
))1
2119891
+(120585119860 +119860120582
2)1
2119891minus120582119891
4(4119892 + 120582)])
(21)
The initial condition is
P119906= 120575 (119909 minus ) for 119891 = 0 (22)
or
P119906= radic
119898120596
ℎ120575 (120585 minus 120585) for 119891 = 0 (23)
For low 119891 = (ℎ1205962) 120573 = ℎ1205962119896119861119879 (high temperature) the
particle should act almost like a free particle as its probablekinetic energy is so high Therefore we expect that for low119891 the density matrix for a harmonic oscillator will be givenapproximately by [7]
P119906119891119903119890
(120585 120585 119891) asymp radic119898120596
4120587ℎ119891119890minus(120585minus
120585)2
4119891
(24)
Comparing the value of 119860 119861 in (21) with (24) identifiesthe solution as
119861
radic2119891= radic
119898120596
4120587ℎ119891997888rarr 119861 = radic
119898120596
2120587ℎ
1205852
minus 2120585 120585 + 1205852
= 1205852
+ (120582 + 2119860) 120585
+ (1198602
+1205822
4+ 119860120582 minus 4119892120582119891
2
minus 1205822
1198912
)
119860 = minus( 120585 +120582
2)
(25)
Substituting (25) into (19) and setting 120585 = 120585 produce
P119906(120585 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus[(120585+(1205822))
2
((cosh(2119891)minus1) sinh(2119891))minus(1205821198914)(120582+4119892)]
(26)
Using the identity of trigonometric tanh(1199092) =
(cosh(119909) minus 1) sinh(119909) and setting 120585 = radic119898120596ℎ119909 into
4 Advances in Physical Chemistry
(26) we can rewrite the canonical density matrix or chargebond-order matrix [8] as
P119906(119909 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus(1198981205964ℎ)[(2119909+radicℎ119898120596120582)
2 tanh(119891)minus(120582119891ℎ119898120596)(120582+4119892)]
(27)
The obtained results are neatly summarized in Figure 1 Itis the probability density for finding the system at 119909 we noteas a Gaussian distribution in 119909 Let us define four new thevariables 120574 120578 120583 and 120601 as function of 119891 by
120574 (119891) = radic119898120596
2120587ℎ sinh (2119891) 120578 (119891) =
119898120596
4ℎtanh (119891)
120583 (119891) =120582119891
4(120582 + 4119892) 120601 = radic
ℎ
119898120596120582
(28)
we can rewrite the density matrix completely in terms of thenew variable (120574 120578 120583 120601) as
P119906(119909 119891) asymp 120574119890
minus(120578(2119909+120601)2
minus120583)
(29)
The expectation value of 1199092 in the density matrix P119906(119909 119891)
can be determined from the definition of the expectationvalues as
⟨1199092
⟩ =intinfin
minusinfin
1199092P119906(119909 119891) 119889119909
intinfin
minusinfin
P119906(119909 119891) 119889119909
=120574radic120587119890
120583
1612057832
2radic120578
120574radic120587119890120583(1 + 2120578120601
2
)
⟨1199092
⟩ =ℎ
2119898120596coth(
ℎ120596120573
2) +
ℎ
41198981205961205822
(30)
The average of the potential energy (119864119904) is thus
⟨119864119904⟩ =
1
21198981205962
⟨1199092
⟩
⟨119864119904⟩ =
ℎ120596
4coth(
ℎ120596120573
2) +
ℎ120596
81205822
(31)
Thekinetic energy per unit length 120591119871(119909 119891) is defined from
the canonical density matrix P119906(119909 119891) as [3]
120591119871(119909 119891) = minus
ℎ2
2119898
1205972
1205971199092P119906(119909 119891) (32)
By substituting (29) into (32) we can write the kinetic energyper unit length completely as follows
120591119871(119909 119879) =
ℎ120596
2tanh( ℎ120596
2119896119861119879)radic
119898120596
2120587ℎ sinh (ℎ120596119896119861119879)
times 119890(120582ℎ1205968119896
119861119879)(120582+4119892)
times [
[
2 minus119898120596
ℎtanh( ℎ120596
2119896119861119879)(2119909 + radic
ℎ
119898120596120582)
2
]
]
times 119890minus(1198981205964ℎ) tanh(ℎ1205962119896
119861119879)(2119909+radicℎ119898120596120582)
2
(33)
The obtained results from (33) are summarized in Figure 2Now the Helmholtz free energy of the harmonic oscilla-
tor asymmetric potential system (119865New) is given by
119865New = minus1
120573lnQ
119873(120573) (34)
or
Q119873(120573) = 119890
minus120573119865New (35)
The density matrix of the system in the integrationrepresentation will be given by
119890minus120573119865new = int
infin
minusinfin
P119906(119909 120573) 119889119909 (36)
Substituting (29) into (36) we obtain for the partitionfunction of the particle in harmonic oscillator asymmetricpotential system
119890minus120573119865new = int
infin
minusinfin
120574119890minus(120578(2119909+120601)
2
minus120583)
119889119909
=120574
2radic120587
120578
=119890(1205821198914)(120582+4119892)
2 sinh (119891)
(37)
Thus we have the following results
119865new =1
120573ln (2 sinh (119891)) minus
120582119891
4120573(120582 + 4119892) (38)
This is the Helmholtz free energy for a one-dimensionalharmonic oscillator asymmetric potential (119865new) as alreadyderived When we write the Helmholtz free energy in newvariable is 119866new = 2119865newℎ120596 119891 = ℎ1205961205732 into (38) we obtainthat
119866new =1
119891ln (2 sinh (119891)) minus 120582
4(120582 + 4119892) (39)
One may summarize the numerical results of Helmholtzfree energy for harmonic oscillator asymmetric potential by
Advances in Physical Chemistry 5
x
05
10
15
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
120588u(x120573)
(a) Density Matrix 120582 = 1 119892 = 1
05
10
15
20
x
155 100minus5minus10minus15
1205821 = 01
1205822 = 20
1205823 = 35
120588u(x120573)
(b) Density Matrix 119892 = 1 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
g1 = 01
g2 = 25
g3 = 55
120588u(x120573)
(c) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
x
155 100minus5minus10minus15
g1 = minus01
g2 = minus45
g3 = minus85
120588u(x120573)
(d) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
1205821 = minus01
1205822 = minus40
1205823 = minus75
120588u(x120573)
(e) Density Matrix 119892 = 1 119879 = 15
Figure 1 Plot of the canonical density matrix of a single particle for harmonic oscillator asymmetric potential system
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
2 Advances in Physical Chemistry
vector that describes a quantum system in a pure state Thedensity matrix is the quantum-mechanical analogue to aphase-space probability measure (probability distribution ofposition and momentum) in classical statistical mechanicsExplicitly suppose a quantum system may be found in state10038161003816100381610038161205951⟩ with probability 119875
1or it may be found in state |120595
2⟩ with
probability1198752or it may be found in state |120595
3⟩with probability
1198753 and so on The density matrix for this system is
P = sum119899
119875119899
1003816100381610038161003816120595119899⟩ ⟨1205951198991003816100381610038161003816 (1)
where (|120595119899⟩) need not be orthogonal and sum
119899119875119899= 1 By
choosing an orthonormal basis (|119906119895⟩) one may resolve the
density operator into the density matrix whose elements are
P119894119895= sum119899
119875119899⟨119906119895| 120595119899⟩ ⟨120595
119899| 119906119894⟩ (2)
For an operator A (which describes an observable A of thesystem) the expectation value ⟨A⟩ is given by
⟨A⟩ = sum119899
119875119899⟨120595119899
10038161003816100381610038161003816A10038161003816100381610038161003816120595119899⟩ = sum
119894
⟨119906119894
10038161003816100381610038161003816PA
10038161003816100381610038161003816119906119894⟩ = tr (PA)
(3)
In the canonical ensemble the probability that the system isin state |119899⟩ is given by
119875119899=
119890minus120573E119899
sum119899119890120573E119899
(4)
and the thermal average of an arbitrary operator A is
⟨A⟩ = sum119899
⟨11989910038161003816100381610038161003816A10038161003816100381610038161003816119899⟩(
119890minus120573E119899
sum119899119890120573E119899
)
=1
Q119873(120573)
sum119899
⟨11989910038161003816100381610038161003816A10038161003816100381610038161003816119899⟩ 119890
minus120573E119899
(5)
The operator
P =1
Q119873(120573)
sum119899
|119899⟩ ⟨119899| 119890minus120573E119899 (6)
is commonly referred to as density operator and its matrixrepresentation as the density matrix Here Q
119873(120573) is the
partition function
3 A Linear Harmonic Oscillator AsymmetricPotential by Feynmans Approach
Next we consider the case of a linear harmonics oscillator ofasymmetric potential whose Hamiltonian is given by [4]
H119878=2
2119898+1
21198981205962
1199092
+ radic119898120596
ℎ
ℎ120596
2120582(119909 minus 119892radic
ℎ
119898120596) (7)
we note that this is a time-independent Hamiltonian ofharmonic oscillator asymmetric axes-shift potential system
Substituting in the time-independent Hamiltonian equation(7) into the Bloch equation [3] we can rewrite the differentialequation completely in terms of P
119906(120585 120585 119891) as
minus120597P119906
120597120573= minus
ℎ2
2119898
1205972P119906
1205971199092+1
21198981205962
1199092
P119906
+ radic119898120596
ℎ
ℎ120596
2120582(119909 minus 119892radic
ℎ
119898120596) P
119906
(8)
We define some new dimensionless variable The positionvariable 119909 = 119909 is replaced with the dimensionless variable120585
120585 = radic119898120596
ℎ119909 119909
2
=ℎ
1198981205961205852
120573 =2119891
ℎ120596 (9)
Substituting in for 119909 in terms of 120585 we can rewrite thedifferential equation completely in terms of 120585 and P
119906as
minus120597P119906
120597119891= minus
1205972P119906
1205971205852+ 1205852
P119906+ 120582120585P
119906minus 120582119892P
119906 (10)
Knowing the Gaussian dependence of the density matrix P119906
in case of the free particle we try the following form
P119906(120585 120585 119891) asymp 119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
120582119892P119906= 120582119892119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
minus120597P119906
120597119891= [120585
2119889119886 (119891)
119889119891+ 120585
119889119887 (119891)
119889119891+119889119888 (119891)
119889119891]
times 119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
minus1205972P
119906
1205971205852= (2119886 (119891) minus 4119886
2
(119891) 1205852
minus 4119886 (119891) 119887 (119891) 120585 minus 1198872
(119891))
times 119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
1205852
P119906= 1205852
119890minus(119886(119891)120585
2
+119887(119891)120585+119888(119891))
120582120585P119906= 120582120585119890
minus(119886(119891)1205852
+119887(119891)120585+119888(119891))
(11)
Substituting expansion equation (11) into (10) leads us to theequation
119889119886 (119891)
1198891198911205852
+119889119887 (119891)
119889119891120585 +
119889119888 (119891)
119889119891
= (1 minus 41198862
(119891)) 1205852
+ (120582 minus 4119886 (119891) 119887 (119891)) 120585
+ (2119886 (119891) minus 120582119892 minus 1198872
(119891))
(12)
Advances in Physical Chemistry 3
Equating the coefficients for 1205852 120585 and 1205850 = 1 on both sides ofthe last equation (12) gives
119889119886 (119891)
119889119891= (1 minus 4119886
2
(119891)) (13)
119889119887 (119891)
119889119891= (120582 minus 4119886 (119891) 119887 (119891)) (14)
119889119888 (119891)
119889119891= (2119886 (119891) minus 120582119892 minus 119887
2
(119891)) (15)
respectively Computing (13) is integrated to give
119886 (119891) =1
2coth (2119891) (16)
Equation (14) is the linear first-order ordinary differentialequation to give [5 6]
120582 =119889119887 (119891)
119889119891+ 2 coth (2119891) 119887 (119891)
119887 (119891) = 119890minusint 2 coth(2119891)119889119891
(int120582119890int 2 coth(2119891)119889119891
119889119891 + 119860)
119887 (119891) =120582
2coth (2119891) + 119860csch (2119891)
(17)
Equation (15) is integrated with respect to 119891 as119889119888 (119891)
119889119891= (coth (2119891) minus 120582119892 minus (120582
2coth (2119891) +119860csch (2119891))
2
)
119888 (119891) = int coth (2119891) 119889119891 minus int120582119892119889119891
minus int(120582
2coth (2119891) + 119860csch (2119891))
2
119889119891
119888 (119891) =1
8(minus2119891120582 (4119892 + 120582) + (4119860
2
+ 1205822
) coth (2119891)
+ 4 ln (sinh (2119891))
+2119860120582csch (119891) sech (119891) minus ln119861) (18)
where 119861 120582 and 119892 are constants Substituting 119886(119891) 119887(119891) and119888(119891) of (16) (17) and (18) respectively into (11) we mayrewrite density matrix asP119906
=119861
radicsinh (2119891)
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
)) coth (2119891)
+(120585119860 +119860120582
2) csch (2119891) minus
119891120582
4(4119892 + 120582) ])
(19)
Thus the Taylor series expansion of sinh(2119891) coth(2119891) about119891 rarr 0 is given by
lim119891rarr0
sinh (2119891) asymp 2119891 lim119891rarr0
coth (2119891) asymp 1
2119891 (20)
we obtain the density matrix final form
P119906asymp
119861
radic2119891
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
))1
2119891
+(120585119860 +119860120582
2)1
2119891minus120582119891
4(4119892 + 120582)])
(21)
The initial condition is
P119906= 120575 (119909 minus ) for 119891 = 0 (22)
or
P119906= radic
119898120596
ℎ120575 (120585 minus 120585) for 119891 = 0 (23)
For low 119891 = (ℎ1205962) 120573 = ℎ1205962119896119861119879 (high temperature) the
particle should act almost like a free particle as its probablekinetic energy is so high Therefore we expect that for low119891 the density matrix for a harmonic oscillator will be givenapproximately by [7]
P119906119891119903119890
(120585 120585 119891) asymp radic119898120596
4120587ℎ119891119890minus(120585minus
120585)2
4119891
(24)
Comparing the value of 119860 119861 in (21) with (24) identifiesthe solution as
119861
radic2119891= radic
119898120596
4120587ℎ119891997888rarr 119861 = radic
119898120596
2120587ℎ
1205852
minus 2120585 120585 + 1205852
= 1205852
+ (120582 + 2119860) 120585
+ (1198602
+1205822
4+ 119860120582 minus 4119892120582119891
2
minus 1205822
1198912
)
119860 = minus( 120585 +120582
2)
(25)
Substituting (25) into (19) and setting 120585 = 120585 produce
P119906(120585 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus[(120585+(1205822))
2
((cosh(2119891)minus1) sinh(2119891))minus(1205821198914)(120582+4119892)]
(26)
Using the identity of trigonometric tanh(1199092) =
(cosh(119909) minus 1) sinh(119909) and setting 120585 = radic119898120596ℎ119909 into
4 Advances in Physical Chemistry
(26) we can rewrite the canonical density matrix or chargebond-order matrix [8] as
P119906(119909 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus(1198981205964ℎ)[(2119909+radicℎ119898120596120582)
2 tanh(119891)minus(120582119891ℎ119898120596)(120582+4119892)]
(27)
The obtained results are neatly summarized in Figure 1 Itis the probability density for finding the system at 119909 we noteas a Gaussian distribution in 119909 Let us define four new thevariables 120574 120578 120583 and 120601 as function of 119891 by
120574 (119891) = radic119898120596
2120587ℎ sinh (2119891) 120578 (119891) =
119898120596
4ℎtanh (119891)
120583 (119891) =120582119891
4(120582 + 4119892) 120601 = radic
ℎ
119898120596120582
(28)
we can rewrite the density matrix completely in terms of thenew variable (120574 120578 120583 120601) as
P119906(119909 119891) asymp 120574119890
minus(120578(2119909+120601)2
minus120583)
(29)
The expectation value of 1199092 in the density matrix P119906(119909 119891)
can be determined from the definition of the expectationvalues as
⟨1199092
⟩ =intinfin
minusinfin
1199092P119906(119909 119891) 119889119909
intinfin
minusinfin
P119906(119909 119891) 119889119909
=120574radic120587119890
120583
1612057832
2radic120578
120574radic120587119890120583(1 + 2120578120601
2
)
⟨1199092
⟩ =ℎ
2119898120596coth(
ℎ120596120573
2) +
ℎ
41198981205961205822
(30)
The average of the potential energy (119864119904) is thus
⟨119864119904⟩ =
1
21198981205962
⟨1199092
⟩
⟨119864119904⟩ =
ℎ120596
4coth(
ℎ120596120573
2) +
ℎ120596
81205822
(31)
Thekinetic energy per unit length 120591119871(119909 119891) is defined from
the canonical density matrix P119906(119909 119891) as [3]
120591119871(119909 119891) = minus
ℎ2
2119898
1205972
1205971199092P119906(119909 119891) (32)
By substituting (29) into (32) we can write the kinetic energyper unit length completely as follows
120591119871(119909 119879) =
ℎ120596
2tanh( ℎ120596
2119896119861119879)radic
119898120596
2120587ℎ sinh (ℎ120596119896119861119879)
times 119890(120582ℎ1205968119896
119861119879)(120582+4119892)
times [
[
2 minus119898120596
ℎtanh( ℎ120596
2119896119861119879)(2119909 + radic
ℎ
119898120596120582)
2
]
]
times 119890minus(1198981205964ℎ) tanh(ℎ1205962119896
119861119879)(2119909+radicℎ119898120596120582)
2
(33)
The obtained results from (33) are summarized in Figure 2Now the Helmholtz free energy of the harmonic oscilla-
tor asymmetric potential system (119865New) is given by
119865New = minus1
120573lnQ
119873(120573) (34)
or
Q119873(120573) = 119890
minus120573119865New (35)
The density matrix of the system in the integrationrepresentation will be given by
119890minus120573119865new = int
infin
minusinfin
P119906(119909 120573) 119889119909 (36)
Substituting (29) into (36) we obtain for the partitionfunction of the particle in harmonic oscillator asymmetricpotential system
119890minus120573119865new = int
infin
minusinfin
120574119890minus(120578(2119909+120601)
2
minus120583)
119889119909
=120574
2radic120587
120578
=119890(1205821198914)(120582+4119892)
2 sinh (119891)
(37)
Thus we have the following results
119865new =1
120573ln (2 sinh (119891)) minus
120582119891
4120573(120582 + 4119892) (38)
This is the Helmholtz free energy for a one-dimensionalharmonic oscillator asymmetric potential (119865new) as alreadyderived When we write the Helmholtz free energy in newvariable is 119866new = 2119865newℎ120596 119891 = ℎ1205961205732 into (38) we obtainthat
119866new =1
119891ln (2 sinh (119891)) minus 120582
4(120582 + 4119892) (39)
One may summarize the numerical results of Helmholtzfree energy for harmonic oscillator asymmetric potential by
Advances in Physical Chemistry 5
x
05
10
15
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
120588u(x120573)
(a) Density Matrix 120582 = 1 119892 = 1
05
10
15
20
x
155 100minus5minus10minus15
1205821 = 01
1205822 = 20
1205823 = 35
120588u(x120573)
(b) Density Matrix 119892 = 1 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
g1 = 01
g2 = 25
g3 = 55
120588u(x120573)
(c) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
x
155 100minus5minus10minus15
g1 = minus01
g2 = minus45
g3 = minus85
120588u(x120573)
(d) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
1205821 = minus01
1205822 = minus40
1205823 = minus75
120588u(x120573)
(e) Density Matrix 119892 = 1 119879 = 15
Figure 1 Plot of the canonical density matrix of a single particle for harmonic oscillator asymmetric potential system
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
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Advances in Physical Chemistry 3
Equating the coefficients for 1205852 120585 and 1205850 = 1 on both sides ofthe last equation (12) gives
119889119886 (119891)
119889119891= (1 minus 4119886
2
(119891)) (13)
119889119887 (119891)
119889119891= (120582 minus 4119886 (119891) 119887 (119891)) (14)
119889119888 (119891)
119889119891= (2119886 (119891) minus 120582119892 minus 119887
2
(119891)) (15)
respectively Computing (13) is integrated to give
119886 (119891) =1
2coth (2119891) (16)
Equation (14) is the linear first-order ordinary differentialequation to give [5 6]
120582 =119889119887 (119891)
119889119891+ 2 coth (2119891) 119887 (119891)
119887 (119891) = 119890minusint 2 coth(2119891)119889119891
(int120582119890int 2 coth(2119891)119889119891
119889119891 + 119860)
119887 (119891) =120582
2coth (2119891) + 119860csch (2119891)
(17)
Equation (15) is integrated with respect to 119891 as119889119888 (119891)
119889119891= (coth (2119891) minus 120582119892 minus (120582
2coth (2119891) +119860csch (2119891))
2
)
119888 (119891) = int coth (2119891) 119889119891 minus int120582119892119889119891
minus int(120582
2coth (2119891) + 119860csch (2119891))
2
119889119891
119888 (119891) =1
8(minus2119891120582 (4119892 + 120582) + (4119860
2
+ 1205822
) coth (2119891)
+ 4 ln (sinh (2119891))
+2119860120582csch (119891) sech (119891) minus ln119861) (18)
where 119861 120582 and 119892 are constants Substituting 119886(119891) 119887(119891) and119888(119891) of (16) (17) and (18) respectively into (11) we mayrewrite density matrix asP119906
=119861
radicsinh (2119891)
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
)) coth (2119891)
+(120585119860 +119860120582
2) csch (2119891) minus
119891120582
4(4119892 + 120582) ])
(19)
Thus the Taylor series expansion of sinh(2119891) coth(2119891) about119891 rarr 0 is given by
lim119891rarr0
sinh (2119891) asymp 2119891 lim119891rarr0
coth (2119891) asymp 1
2119891 (20)
we obtain the density matrix final form
P119906asymp
119861
radic2119891
times exp(minus[(1205852
2+120585120582
2+1
8(4119860
2
+ 1205822
))1
2119891
+(120585119860 +119860120582
2)1
2119891minus120582119891
4(4119892 + 120582)])
(21)
The initial condition is
P119906= 120575 (119909 minus ) for 119891 = 0 (22)
or
P119906= radic
119898120596
ℎ120575 (120585 minus 120585) for 119891 = 0 (23)
For low 119891 = (ℎ1205962) 120573 = ℎ1205962119896119861119879 (high temperature) the
particle should act almost like a free particle as its probablekinetic energy is so high Therefore we expect that for low119891 the density matrix for a harmonic oscillator will be givenapproximately by [7]
P119906119891119903119890
(120585 120585 119891) asymp radic119898120596
4120587ℎ119891119890minus(120585minus
120585)2
4119891
(24)
Comparing the value of 119860 119861 in (21) with (24) identifiesthe solution as
119861
radic2119891= radic
119898120596
4120587ℎ119891997888rarr 119861 = radic
119898120596
2120587ℎ
1205852
minus 2120585 120585 + 1205852
= 1205852
+ (120582 + 2119860) 120585
+ (1198602
+1205822
4+ 119860120582 minus 4119892120582119891
2
minus 1205822
1198912
)
119860 = minus( 120585 +120582
2)
(25)
Substituting (25) into (19) and setting 120585 = 120585 produce
P119906(120585 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus[(120585+(1205822))
2
((cosh(2119891)minus1) sinh(2119891))minus(1205821198914)(120582+4119892)]
(26)
Using the identity of trigonometric tanh(1199092) =
(cosh(119909) minus 1) sinh(119909) and setting 120585 = radic119898120596ℎ119909 into
4 Advances in Physical Chemistry
(26) we can rewrite the canonical density matrix or chargebond-order matrix [8] as
P119906(119909 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus(1198981205964ℎ)[(2119909+radicℎ119898120596120582)
2 tanh(119891)minus(120582119891ℎ119898120596)(120582+4119892)]
(27)
The obtained results are neatly summarized in Figure 1 Itis the probability density for finding the system at 119909 we noteas a Gaussian distribution in 119909 Let us define four new thevariables 120574 120578 120583 and 120601 as function of 119891 by
120574 (119891) = radic119898120596
2120587ℎ sinh (2119891) 120578 (119891) =
119898120596
4ℎtanh (119891)
120583 (119891) =120582119891
4(120582 + 4119892) 120601 = radic
ℎ
119898120596120582
(28)
we can rewrite the density matrix completely in terms of thenew variable (120574 120578 120583 120601) as
P119906(119909 119891) asymp 120574119890
minus(120578(2119909+120601)2
minus120583)
(29)
The expectation value of 1199092 in the density matrix P119906(119909 119891)
can be determined from the definition of the expectationvalues as
⟨1199092
⟩ =intinfin
minusinfin
1199092P119906(119909 119891) 119889119909
intinfin
minusinfin
P119906(119909 119891) 119889119909
=120574radic120587119890
120583
1612057832
2radic120578
120574radic120587119890120583(1 + 2120578120601
2
)
⟨1199092
⟩ =ℎ
2119898120596coth(
ℎ120596120573
2) +
ℎ
41198981205961205822
(30)
The average of the potential energy (119864119904) is thus
⟨119864119904⟩ =
1
21198981205962
⟨1199092
⟩
⟨119864119904⟩ =
ℎ120596
4coth(
ℎ120596120573
2) +
ℎ120596
81205822
(31)
Thekinetic energy per unit length 120591119871(119909 119891) is defined from
the canonical density matrix P119906(119909 119891) as [3]
120591119871(119909 119891) = minus
ℎ2
2119898
1205972
1205971199092P119906(119909 119891) (32)
By substituting (29) into (32) we can write the kinetic energyper unit length completely as follows
120591119871(119909 119879) =
ℎ120596
2tanh( ℎ120596
2119896119861119879)radic
119898120596
2120587ℎ sinh (ℎ120596119896119861119879)
times 119890(120582ℎ1205968119896
119861119879)(120582+4119892)
times [
[
2 minus119898120596
ℎtanh( ℎ120596
2119896119861119879)(2119909 + radic
ℎ
119898120596120582)
2
]
]
times 119890minus(1198981205964ℎ) tanh(ℎ1205962119896
119861119879)(2119909+radicℎ119898120596120582)
2
(33)
The obtained results from (33) are summarized in Figure 2Now the Helmholtz free energy of the harmonic oscilla-
tor asymmetric potential system (119865New) is given by
119865New = minus1
120573lnQ
119873(120573) (34)
or
Q119873(120573) = 119890
minus120573119865New (35)
The density matrix of the system in the integrationrepresentation will be given by
119890minus120573119865new = int
infin
minusinfin
P119906(119909 120573) 119889119909 (36)
Substituting (29) into (36) we obtain for the partitionfunction of the particle in harmonic oscillator asymmetricpotential system
119890minus120573119865new = int
infin
minusinfin
120574119890minus(120578(2119909+120601)
2
minus120583)
119889119909
=120574
2radic120587
120578
=119890(1205821198914)(120582+4119892)
2 sinh (119891)
(37)
Thus we have the following results
119865new =1
120573ln (2 sinh (119891)) minus
120582119891
4120573(120582 + 4119892) (38)
This is the Helmholtz free energy for a one-dimensionalharmonic oscillator asymmetric potential (119865new) as alreadyderived When we write the Helmholtz free energy in newvariable is 119866new = 2119865newℎ120596 119891 = ℎ1205961205732 into (38) we obtainthat
119866new =1
119891ln (2 sinh (119891)) minus 120582
4(120582 + 4119892) (39)
One may summarize the numerical results of Helmholtzfree energy for harmonic oscillator asymmetric potential by
Advances in Physical Chemistry 5
x
05
10
15
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
120588u(x120573)
(a) Density Matrix 120582 = 1 119892 = 1
05
10
15
20
x
155 100minus5minus10minus15
1205821 = 01
1205822 = 20
1205823 = 35
120588u(x120573)
(b) Density Matrix 119892 = 1 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
g1 = 01
g2 = 25
g3 = 55
120588u(x120573)
(c) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
x
155 100minus5minus10minus15
g1 = minus01
g2 = minus45
g3 = minus85
120588u(x120573)
(d) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
1205821 = minus01
1205822 = minus40
1205823 = minus75
120588u(x120573)
(e) Density Matrix 119892 = 1 119879 = 15
Figure 1 Plot of the canonical density matrix of a single particle for harmonic oscillator asymmetric potential system
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
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4 Advances in Physical Chemistry
(26) we can rewrite the canonical density matrix or chargebond-order matrix [8] as
P119906(119909 119891) asymp radic
119898120596
2120587ℎ sinh (2119891)
times 119890minus(1198981205964ℎ)[(2119909+radicℎ119898120596120582)
2 tanh(119891)minus(120582119891ℎ119898120596)(120582+4119892)]
(27)
The obtained results are neatly summarized in Figure 1 Itis the probability density for finding the system at 119909 we noteas a Gaussian distribution in 119909 Let us define four new thevariables 120574 120578 120583 and 120601 as function of 119891 by
120574 (119891) = radic119898120596
2120587ℎ sinh (2119891) 120578 (119891) =
119898120596
4ℎtanh (119891)
120583 (119891) =120582119891
4(120582 + 4119892) 120601 = radic
ℎ
119898120596120582
(28)
we can rewrite the density matrix completely in terms of thenew variable (120574 120578 120583 120601) as
P119906(119909 119891) asymp 120574119890
minus(120578(2119909+120601)2
minus120583)
(29)
The expectation value of 1199092 in the density matrix P119906(119909 119891)
can be determined from the definition of the expectationvalues as
⟨1199092
⟩ =intinfin
minusinfin
1199092P119906(119909 119891) 119889119909
intinfin
minusinfin
P119906(119909 119891) 119889119909
=120574radic120587119890
120583
1612057832
2radic120578
120574radic120587119890120583(1 + 2120578120601
2
)
⟨1199092
⟩ =ℎ
2119898120596coth(
ℎ120596120573
2) +
ℎ
41198981205961205822
(30)
The average of the potential energy (119864119904) is thus
⟨119864119904⟩ =
1
21198981205962
⟨1199092
⟩
⟨119864119904⟩ =
ℎ120596
4coth(
ℎ120596120573
2) +
ℎ120596
81205822
(31)
Thekinetic energy per unit length 120591119871(119909 119891) is defined from
the canonical density matrix P119906(119909 119891) as [3]
120591119871(119909 119891) = minus
ℎ2
2119898
1205972
1205971199092P119906(119909 119891) (32)
By substituting (29) into (32) we can write the kinetic energyper unit length completely as follows
120591119871(119909 119879) =
ℎ120596
2tanh( ℎ120596
2119896119861119879)radic
119898120596
2120587ℎ sinh (ℎ120596119896119861119879)
times 119890(120582ℎ1205968119896
119861119879)(120582+4119892)
times [
[
2 minus119898120596
ℎtanh( ℎ120596
2119896119861119879)(2119909 + radic
ℎ
119898120596120582)
2
]
]
times 119890minus(1198981205964ℎ) tanh(ℎ1205962119896
119861119879)(2119909+radicℎ119898120596120582)
2
(33)
The obtained results from (33) are summarized in Figure 2Now the Helmholtz free energy of the harmonic oscilla-
tor asymmetric potential system (119865New) is given by
119865New = minus1
120573lnQ
119873(120573) (34)
or
Q119873(120573) = 119890
minus120573119865New (35)
The density matrix of the system in the integrationrepresentation will be given by
119890minus120573119865new = int
infin
minusinfin
P119906(119909 120573) 119889119909 (36)
Substituting (29) into (36) we obtain for the partitionfunction of the particle in harmonic oscillator asymmetricpotential system
119890minus120573119865new = int
infin
minusinfin
120574119890minus(120578(2119909+120601)
2
minus120583)
119889119909
=120574
2radic120587
120578
=119890(1205821198914)(120582+4119892)
2 sinh (119891)
(37)
Thus we have the following results
119865new =1
120573ln (2 sinh (119891)) minus
120582119891
4120573(120582 + 4119892) (38)
This is the Helmholtz free energy for a one-dimensionalharmonic oscillator asymmetric potential (119865new) as alreadyderived When we write the Helmholtz free energy in newvariable is 119866new = 2119865newℎ120596 119891 = ℎ1205961205732 into (38) we obtainthat
119866new =1
119891ln (2 sinh (119891)) minus 120582
4(120582 + 4119892) (39)
One may summarize the numerical results of Helmholtzfree energy for harmonic oscillator asymmetric potential by
Advances in Physical Chemistry 5
x
05
10
15
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
120588u(x120573)
(a) Density Matrix 120582 = 1 119892 = 1
05
10
15
20
x
155 100minus5minus10minus15
1205821 = 01
1205822 = 20
1205823 = 35
120588u(x120573)
(b) Density Matrix 119892 = 1 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
g1 = 01
g2 = 25
g3 = 55
120588u(x120573)
(c) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
x
155 100minus5minus10minus15
g1 = minus01
g2 = minus45
g3 = minus85
120588u(x120573)
(d) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
1205821 = minus01
1205822 = minus40
1205823 = minus75
120588u(x120573)
(e) Density Matrix 119892 = 1 119879 = 15
Figure 1 Plot of the canonical density matrix of a single particle for harmonic oscillator asymmetric potential system
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
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CatalystsJournal of
Advances in Physical Chemistry 5
x
05
10
15
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
120588u(x120573)
(a) Density Matrix 120582 = 1 119892 = 1
05
10
15
20
x
155 100minus5minus10minus15
1205821 = 01
1205822 = 20
1205823 = 35
120588u(x120573)
(b) Density Matrix 119892 = 1 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
g1 = 01
g2 = 25
g3 = 55
120588u(x120573)
(c) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
x
155 100minus5minus10minus15
g1 = minus01
g2 = minus45
g3 = minus85
120588u(x120573)
(d) Density Matrix 120582 = 1 119892 = + 119879 = 15
05
10
15
20
x
155 100minus5minus10minus15
1205821 = minus01
1205822 = minus40
1205823 = minus75
120588u(x120573)
(e) Density Matrix 119892 = 1 119879 = 15
Figure 1 Plot of the canonical density matrix of a single particle for harmonic oscillator asymmetric potential system
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Carbohydrate Chemistry
International Journal of
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Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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Analytical Methods in Chemistry
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Medicinal ChemistryInternational Journal of
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Chromatography Research International
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CatalystsJournal of
6 Advances in Physical Chemistry
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
1205821 = 01
1205822 = 10
1205823 = 19
(a) Kinetic Energy 119892 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
1205821 = minus01
1205822 = minus16
1205823 = minus28
x
155 100minus5minus10minus15
(b) Kinetic Energy 119892 = 1 119879 = 5K
x
155 100minus5minus10minus15
002
004
006
008
010
012
120591L(xT
)
g1 = 01
g2 = 16
g3 = 28
(c) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
g1 = minus01
g2 = minus16
g3 = minus28
x
155 100minus5minus10minus15
(d) Kinetic Energy 120582 = 1 119879 = 5K
002
004
006
008
010
120591L(xT
)
x
T1 = 5KT2 = 10KT3 = 15K
155 100minus5minus10minus15
(e) Kinetic Energy 120582 = 1 119892 = 1
Figure 2 Illustration the kinetic energy per unit length under harmonic oscillator asymmetric potential system
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Advances in Physical Chemistry 7
0 2 4 6 8 100
1
2
3
4
5
1205821 = 01
1205822 = 10
1205823 = 60
T (K)
S new(T
)
(a) Entropy (119878new(119879)) 119892 = minus2
0
1
2
3
4
5
0 2 4 6 8 10T (K)
minus1
1205821 = minus001
12058221205823
= minus003
= minus005
S new(T
)
(b) Entropy (119878new(119879)) 119892 = minus4
0
2
4
0 2 4 6 8 10T (K)
minus2
minus4
g1 = 010
g2 = 350
g3 = 670
S new(T
)
(c) Entropy (119878new(119879)) 120582 = 005
0
1
2
3
4
5
S new
(T)
0 2 4 6 8 10T (K)
g1 = minus010
g2 = minus1850
g3 = minus4270
(d) Entropy (119878new(119879)) 120582 = 005
Figure 3 Illustration of the entropy(119878) of a single particle under harmonic oscillator asymmetric potential
Feynmans approach in Section 5 In Section 5 comparisonof Helmholtz free energy for harmonic oscillator asymmetricpotential obtain from Feynmans approach and the path-integral method 120581(119910) This work aims at computing theentropy (119878) of the harmonic oscillator asymmetric potentialusing the Helmholtz free energy in (38) Using the relation
119878new (119879) = minus120597119865new (119879)
120597119879= 119896119861(120597
120597119879(119879 ln (Q
119873(120573)))) (40)
the entropy of a system can be write as
119878new (119879) =119896119861ℎ120596
4 (119896119861119879)
cosh (ℎ1205962119896119861119879)
sinh (ℎ1205962119896119861119879)
minus 119896119861ln (2 sinh (ℎ1205962119896
119861119879))
minus119896119861120582120596ℎ
8 (119896119861119879)2(120582 + 4119892)
(41)
Equation (41) is plotted in Figure 3(a) assuming the valueof axes-shift potential 119892 equal minus2 and vary the value ofasymmetric potential 120582 From (41) are plotted in Figure 3assuming the value of asymmetric potential 120582 equal 005 andvary the value of axes-shift potential 119892 Figure 3 shows thatas 120582 increases the depth of the entropy decreases and itsminimummoves farther away the origin
4 Evaluate Helmholtz Free Energy via Path-Integral Method (120581(119910))
In the Helmholtz free energy the quantity of the thermody-namics of a given harmonic oscillator asymmetric potentialsystem is derived from its the path-integral method
120581 (119910) = radic6119898119896
119861119879
120587ℎ2intinfin
minusinfin
119890minus(6119898119896
119861119879(119910minus119911)
2
ℎ2
)
119881 (119911) 119889119911 (42)
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
8 Advances in Physical Chemistry
In this case of the harmonic oscillator asymmetric potentialis
119881 (119911) =1
21198961199112
+ 120591120582 (119911 minus 120593119892) (43)
where setting 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962) 120593 = radicℎ119898120596
and substituting (43) into (42) we can write classical 120581(119910) tosimply produce
120581 (119910) = Φintinfin
minusinfin
(1
21198961199112
+ 120591120582 (119911 minus 120593119892)) 119890minus120572(119911minus119910)
2
119889119911 (44)
where 120572 = 6119898120573ℎ2 Φ = radic6119898120573120587ℎ2 The final the path-integral method is
120581 (119910) =Φradic120587 (119896 + 2119896119910
2
120572 + 4120572120582120591 (119910 minus 120593119892))
412057232 (45)
In such case it is more useful to write the partition functionof 120581(119910) as
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573120581(119910)
119889119910 (46)
where Ω = 1198982120587ℎ119880 Substituting (45) into (46) can obtainthe partition function of path-integral method (120581(119910)) in caseof harmonic oscillator asymmetric potential
119890minus120573119865119888119897minus119870 = Ωint
infin
minusinfin
119890minus120573(Φradic120587(119896+2119896119910
2
120572+4120572120582120591(119910minus120593119892))412057232
)
119889119910
= Ω119890minus((120573radic120587Φ1198964120572
32
)minus(4119892120573120572120591120582120593412057232
))
times intinfin
minusinfin
119890minus120573radic120587Φ((2119896120572119910
2
+4120572120591120582119910)412057232
)
119889119910
119890minus120573119865119888119897minus119870 =
radic2 4radic120587Ω
radic120573119896Φradic120572
[119890minus(120573Φradic1205874119896120572
32
)(1198962
minus21205721205822
1205912
minus4119892119896120572120582120591120593)
]
(47)
Substituting 120572 = 6119898120573ℎ2 119896 = 1198981205962 120591 = radic119898120596ℎ (ℎ1205962)120593 = radicℎ119898120596 Φ = radic6119898120573120587ℎ2 and Ω = radic1198982120587ℎ119880 into (47)leads to
119890minus120573119865119888119897minus119870
=1
ℎ120573120596
times 119890minus(1205732
ℎ2
241198982
1205962
)(1198982
1205964
minus(31205822
1198982
1205963
120573ℎ)minus(121198921205821198982
1205962
120573ℎ))
minus120573119865119888119897minus119870
= minus ln (2119891) minus (1198912
6minus1198911205822
4minus 119892119891120582)
(48)
Accordingly the Helmholtz free energy the path-integralmethod (120581(119910)) is given by
119865119888119897minus119870
(119891) =1
120573[ln (2119891) + (
1198912
6minus1198911205822
4minus 119892119891120582)] (49)
Table 1 Comparison of the Helmholtz free energy of the harmonicoscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where 120582 is the value of
asymmetric potential 119892 is the value of axes-shift potential systemand 119879 is the temperature setting 119892 = 120582 = 1
119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Difference1 minus1167350290774 minus1166666666667 00585791892 minus3981008518269 minus3980922055573 00021719023 minus7813921629128 minus7813895954231 00003285794 minus12319532395568 minus12319521555626 00000879895 minus17327718009705 minus17327712457674 00000320416 minus22737227955451 minus22737224741848 00000141337 minus28480839348833 minus28480837324869 710640 times 10minus6
8 minus34510649356211 minus34510648000211 392922 times 10minus6
9 minus40790784085205 minus40790783132793 233487 times 10minus6
10 minus47293369220882 minus47293368526548 146814 times 10minus6
11 minus53996120765666 minus53996120243988 966139 times 10minus7
12 minus60880815552301 minus60880815150468 660032 times 10minus7
13 minus67932273353648 minus67932273053847 465256 times 10minus7
14 minus75137653101331 minus75137652848275 336789 times 10minus7
15 minus82485950683258 minus82485950477511 249433 times 10minus7
After introducing new variable of the Helmholtz free energypath-integral method (120581(119910)) as 119866
119888119897minus119870= 2119865
119888119897minus119870ℎ120596 119891 =
ℎ1205961205732 the Helmholtz free energy of the path-integralmethod (120581(119910)) is then transformed to
119866119888119897minus119870
(119891) =1
119891ln (2119891) +
119891
6minus1205822
4minus 119892120582 (50)
This is known as the Helmholtz free energy of the path-integral method (120581(119910)) We can calculate numerical of theHelmholtz free energy Feynmans (119866new(119891) approach inSection 3) and the Helmholtz free energy of the path-integralmethod (120581(119910))(119866
119888119897minus119870(119891) in Section 4) for the harmonic oscil-
lator asymmetric potential by mathematica program to giveTables 1 and 2 as shown in Section 5
5 The Numerical ResultIn this section we have evaluated numerical of theHelmholtzfree energy Feynmans (119866new(119891)) approach and theHelmholtzfree energy of the path-integral method (120581(119910))(119866
119888119897minus119870(119891)) for
the harmonic oscillator asymmetric potential is (39) and (50)By substituting 119891 = ℎ1205962119896
119861119879 into (39) and (50) we obtain
the Helmholtz free energy Feynmans (119866new(119879)) approach asfollows
119866new (119879) =2119896119861119879
ℎ120596ln(2 sinh( ℎ120596
2119896119861119879)) minus
120582
4(120582 + 4119892) (51)
and the Helmholtz free energy of the path-integral method(120581(119910))(119866
119888119897minus119870(119879))
119866119888119897minus119870
(119879) =2119896119861119879
ℎ120596ln( ℎ120596
119896119861119879) +
ℎ120596
12119896119861119879minus1205822
4minus 119892120582 (52)
For the beginning of the numerical shootingmethodwe needto input the parameters 120582 119892 and 119879 and (39) and (50) into
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Advances in Physical Chemistry 9
Table 2 Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach(119866new) path-integral method 120581(119910)(119866
119888119897minus119870(119879)) where we may vary the value of 119892 120582 119879
120582 119892 119879(119870) 119866new(119879) 119866119888119897minus119870
(119879) Differenceminus1 1 1 0832649709225 0833333333333 0082068555minus2 2 2 0268991481731 0269077944427 0032138119minus3 3 3 0186078370872 0186104045769 0013796943minus4 4 4 0930467604432 0930478444374 0001164993minus5 5 5 2672281990295 2672287542326 0000207763minus6 6 6 5512772044548 5512775258152 0000058294minus7 7 7 9519160651167 9519162675130 0000021262minus8 8 8 14739350643789 14739351999789 9199869 times 10
minus6
minus9 9 9 21209215914795 21209216867207 4490557 times 10minus6
minus10 10 10 28956630779118 28956631473452 2397841 times 10minus6
minus11 11 11 38003879234334 38003879756012 1372696 times 10minus6
minus12 12 12 48369184447699 48369184849532 8307624 times 10minus7
minus13 13 13 60067726646352 60067726962410 5261694 times 10minus7
minus14 14 14 73112346898668 73112347151725 3461208 times 10minus7
minus15 15 15 87514049316743 87514049522489 2351017 times 10minus7
themathematical program [9] Next we obtain the numericalvalues of 119866new(119879) and 119866119888119897minus119870(119879) for the harmonic oscillatorasymmetric potential as displayed in Tables 1 and 2
From Table 1 we show that magnitude of the Helmholtzfree energy decrease from minus11673 to minus824859 with increas-ing the temperature of system From Table 2 we illustratemagnitude of theHelmholtz free energy increase from 08326
to 875140 with increasing the temperature of systemNevertheless 119866
119888119897minus119870(119879) is not as useful as Tables 1 and 2
might indicate The first it can not be used when quantummechanics exchange effect exist The second it fails in itspresent form when the potential 119881(119910) has a very largederivative as in the case of hard-sphere interatomic potential(see Lennard-Jones potential [10])
6 ConclusionThiswork is interested in finding the canonical densitymatrixby Feynmanrsquos technique and kinetic energy per unit lengthand evaluation of theHelmholtz free energyThe results showthat themagnitude of the densitymatrix (P
119906) and the kinetic
energy per unit length (120591119871) vary according to the parameters
120582 119892 and 119879 respectively The magnitude of (P119906) and (120591
119871)
for these cases 120582 is positive with the graph of (P119906) and (120591
119871)
moving from right hand side to left hand side Moreover wefound that as the values of119892have increased and the amplitudeof (P
119906) and (120591
119871) has decreased
Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper
AcknowledgmentsPiyarut Moonsri and Artit Hutem wish to thank the InstituteResearch and Development Chemistry Division and Physics
Division Faculty of Science and Technology PhetchabunRajabhat University Thailand This work is supported by theInstitute Research and Development Phetchabun RajabhatUniversity
References
[1] N H March and A M Murray ldquoRelation between Dirac andcanonical density matrices with applications to imperfectionsin metalsrdquo Physical Review Letters vol 120 no 3 pp 830ndash8361960
[2] I A Howard A Minguzzi N H March and M Tosi ldquoSlatersum for the one-dimensional sech2 potential in relation to thekinetic energy densityrdquo Journal of Mathematical Physics vol 45no 6 pp 2411ndash2419 2004
[3] N H March and L M Nieto ldquoBloch equation for the canonicaldensity matrix in terms of its diagonal element the Slater sumrdquoPhysics Letters A General Atomic and Solid State Physics vol373 no 18-19 pp 1691ndash1692 2009
[4] R K Pathria Statistical Mechanics Butterworth HeinemannOxford UK 1996
[5] G B Arfken and H J Weber Mathematical Methods forPhysicists Elsevier Academic Press New York NY USA 6thedition 2005
[6] K F Riley and M P HobsonMathematical Method for Physicsand Engineering CambridgeUniversity Press 3rd edition 2006
[7] R P Feynman Statistical Mechanics Addison-Wesley LondonUK 1972
[8] I N Levin Quantum Chemistry Pearson Prentice Hall 6thedition 2009
[9] R L Zimmerman and F I Olness Mathematica for PhysicsAddison-Wesley Reading Mass USA 2nd edition 2002
[10] A Hutem and S Boonchui ldquoNumerical evaluation of secondand third virial coefficients of some inert gases via classicalcluster expansionrdquo Journal of Mathematical Chemistry vol 50no 5 pp 1262ndash1276 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of