A gauge-origin independent expression for the Verdet constant within the time-dependent density...

A Gauge-Origin Independent Expressionfor the Verdet Constant within theTime-Dependent Density FunctionalTheory

ARUP BANERJEE,1* JOCHEN AUTSCHBACH,2 TOM ZIEGLER1

1Department of Chemistry, University of Calgary, University Drive NW 2500, Calgary, Alberta,Canada T2N 1N42Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-300

Received 25 September 2003; accepted 2 April 2004Published online 13 October 2004 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20313

ABSTRACT: The analytical expression for the Verdet constant of magneto-opticalrotation is presented within the time-dependent density functional theory formalism.Use has been made of gauge, including atomic orbitals, to circumvent the problem ofgauge dependence often associated with the calculation of magnetic properties. © 2004Wiley Periodicals, Inc. Int J Quantum Chem 101: 572–578, 2005

Key words: Verdet constant; time-dependent density functional theory; responsetheory; magneto-optical rotation

I. Introduction

T he plane of polarization of a linearly polarizedlight undergoes a rotation on passing through

any substance placed in a static magnetic field witha nonzero component along the propagation direc-tion of the light beam. This effect was first observedby Faraday in 1845 [1, 2] and is termed the magne-to-optical activity (MOA) or the Faraday effect. Therotation � of the plane of polarization was later

quantified by Verdet [3]. He found that, in theregion far from absorption, the angle of rotation canbe expressed as

� � V�� B�, (1)

where � is the rotation per unit length of the sam-ple, B� is the magnitude of the magnetic field alongthe direction of propagation of light, and the pro-portionality factor V(�) is the Verdet constant. TheVerdet constant is a material property that dependson the frequency of the incident light. Physically,both aspects can be understood in a unified way interms of the differential behavior of the right (R)

Correspondence to: T. Ziegler; e-mail: [email protected]*Present address: Laser Programme, Centre For Advanced

Technology, Indore–452013, India.

International Journal of Quantum Chemistry, Vol 101, 572–578 (2005)© 2004 Wiley Periodicals, Inc.

and left (L) circular polarized components of inci-dent light in the presence of a symmetry-breakingmagnetic field. In other words, the rotation � iscaused by the different refractive indices being seenby the left and the right circular polarized compo-nents. In particular, the rotation is proportional tothe difference nL � nR (where nL and nR are therefractive indices of the left and the right circularpolarized components, respectively). In the vicinityof absorption (that is, near resonance) the rotationof the plane of polarization is accompanied by thedevelopment of ellipticity (�) [4]. In this study wefocus our attention on the nonabsorptive region.

It has already been shown that [4–6] the rotation� is proportional to the change in the imaginarypart of the frequency-dependent linear polarizabil-ity ��(�) due to the application of a static magneticfield. As a result of this, the Verdet constant can beexpressed as

V�� C��i ��

B��

B��0

, (2)

with ��i (�) and �� denoting the imaginary part of

the frequency-dependent linear polarizability andthe third-rank Levi–Civita tensor, respectively, andthe constant C in SI units is given by

C �26 � 1

2c��2N4�0

� , (3)

where �0 is the permittivity of free space and N isnumber of molecules per unit volume. In this articlewe present an analytical expression for the first-order derivative ��(�) with respect to the mag-netic field B� within the time-dependent densityfunctional theory (TDDFT). This analytical expres-sion can then be used to evaluate the Verdet con-stant via Eq. (2).

Before we begin our discussion we note that inthe past the Verdet constant has been computed atvarious levels of ab initio theory [7–12]. In Ref. [9]the Verdet constant of the H2 and D2 molecules wasobtained from a sum-over state expression of third-order hyperpolarizabilities with correlated wavefunctions. On the other hand, Ref. [10] calculatedthe Verdet constant as a quadratic dipole-dipole-magnetic dipole response function with a self-con-sistent field (SCF) wave function. Using the samequadratic response function approach, it has beenpossible to obtain V(�) within the multiconfigura-

tion SCF [11] and coupled cluster [12] frameworks,respectively.

In the following we derive an analytic expressionfor ��

i (�)/B� within the TDDFT formalism thatsatisfies the (2n � 1) rule of perturbation theory. Itis appropriate to mention here that our method ofderivation is similar to the one described in Refs.[18, 19]. In particular our result is closer to that ofQuinet and Champagne [19], who derived an ana-lytical expression for the derivative of the frequency-dependent polarizability with respect to atomic co-ordinates.

The article is organized as follows. In Section IIwe describe the derivation of TDDFT-based expres-sion of ��

i (�)/B�. We conclude with a discussionin Section III.

II. Theory

We begin with the expression for the frequency-dependent linear polarizability as given by

�� Tr�Hext� D��, (4)

where Hext� is the matrix of the �-th component of

the dipole moment operator in the basis of atomicorbitals (AO), D�(�) is the first-order derivative ofthe density matrix with respect to the component ofelectric field directed along the � axis (where � � x,y, z), and oscillating at a frequency �. We write thederivative of linear polarizability as

��

B�

� � Tr�Hext

� D��

B�

� �Tr�Hext��D�� Hext

� D��, (5)

where D��(�) is the mixed derivative of the densitymatrix with respect to the electric and the magneticfield along the � and the � directions, respectively.In the following we derive an expression for��(�) by using the above relation and the coupledperturbed time-dependent Kohn–Sham (KS) equa-tions. The general form of the time-dependent KSequations (in atomic units) is given by

��12 �2 � veff�r, t� � i

t��i�r, t� � �j

�ij�t��i�r, t�,

(6)

A GAUGE-ORIGIN INDEPENDENT EXPRESSION

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 573

where veff(r, t) is the time-dependent KS potential,and the time-dependent density (r, t) is obtainedfrom the occupied orbitals by

�r, t� � �i

occ

�*i �r, t��i�r, t�. (7)

The orbitals �i(r, t) are expanded in terms of atime-independent basis set of AOs {��} as

�i�r, t� � ��

��r�C�i�t�. (8)

With this the time-dependent KS equations can beput in a matrix form given by

FsC � ��SC�

t � SC�, (9)

where Fs is the AO matrix element of the operator[�(1/2)�2 � veff(r, t)], and S is the overlap matrix ofthe AOs given by

S�� *��r��r�dr. (10)

In the matrix notation the orthogonality relationand the density matrix D read as

�C†SC�

t � 0 (11)

and

D � CnC†, (12)

respectively. In the above equation n denotes theoccupation number matrix. For closed-shell sys-tems it is diagonal with components either equal to2 (occupied KS orbitals) or 0 (virtual KS orbitals).Now we proceed with Eqs. (9)–(12) to develop aperturbative formalism. We consider that the sys-tem is perturbed by three applied potentials and thetotal perturbation Hamiltonian is given by

Hpert�t� � �Hext�100��t� � �Hext

�010��t� � �Hext�001�. (13)

For our purpose Hext(100)(t) and Hext

(010)(t) represent thetime-dependent Hamiltonian due to the appliedelectric fields along the � and � directions, respec-

tively, and Hext(001) denotes the Hamiltonian for the

static external magnetic field along the � direction.These are given by

�Hext�100��t� � ��r�cos �t

�Hext�010��t� � ��r�cos �t

�Hext�001� � ��r� � p��. (14)

At this point it is important to mention that thecalculation of magnetic properties is associatedwith the so-called gauge problem. The problemarises because of the dependence of the results onthe choice of the gauge origin for the vector poten-tial A of the magnetic field, unless use has beenmade of a complete set of atomic orbitals. A num-ber of methods are available in the literature to dealwith such an unphysical feature [13–16]. Here wemake use of the gauge including atomic orbitals(GIAOs) [15, 16] to circumvent the gauge originproblem. In accordance with this approach, theAOs are chosen to be [17]

��r� � exp��B� � R� �� r��r�, (15)

where B� is the static magnetic field vector, and R� � isthe position vector of the atom where the field freeAO ��(r) is centered. To develop the perturbationformalism we expand F, C, �, ��(r), and S as

F�t� � �mnk

�m�n�kF�mnk��t�

C�t� � �mnk

�m�n�kC�mnk��t�

��t� � �mnk

�m�n�k��mnk��t�

��r� � �k

�k��00k��r�

S � �k

�kS�00k�, (16)

and write the time-dependent perturbed matricesby following relations

C�100��t� � C�100��1�e��t � C�100��1�e��t

C�010��t� � C�010��1�e��t � C�010��1�e��t

C�001��t� � C�001� � �0�

C�011��t� � C�011��1�e��t � C�011��1�e��t. (17)

BANERJEE, AUTSCHBACH, AND ZIEGLER

574 VOL. 101, NO. 5

In the above equations the arguments 1 of the Cmatrix denote the frequency �; this notation isfollowed throughout this article for other time-de-pendent quantities also. In a similar fashion F(t)and �(t) are resolved. In the following we denoteS(000) by S for simplicity of notation. By substitutingEqs. (16) and (17) in Eq. (9) we derive the equationsfor the perturbed orbitals. The relevant equationsfor our purpose are:

▪ zeroth order in ��

F�000�C�000� � SC�000��000� (18)▪ First order in �

F�000�C�100��1� � F�100��1�C�000� � �SC�100��1�

� SC�000��100��1� � SC�100��1��000� (19)▪ First order in �

F�000�C�010��1� � F�010��1�C�000� � �SC�010��1�

� SC�000��010��1� � SC�010��1��000� (20)▪ First order in �

F�000�C�001��0� � F�001��0�C�000� � SC�000��001��0�

� SC�001��0��000� � S�001�C�000��000� (21)▪ of order ��

F�000�C�011��1� � F�010��1�C�001��0�

� F�001��0�C�010��1� � F�011��1�C�000�

� �SC�011��1� � �S�001�C�010��1� � SC�000��011�

� �1� � SC�010��1��001��0� � SC�001��0��010�

� �1� � SC�011��1��000� � S�001�C�010��1��000�

� S�001�C�000��010��1� (22)

The normalization condition Eq. (11) at differentorders reads

▪ zeroth order in ��

C†�000�SC�000� � 1 (23)▪ First order in �

C†�000�SC�100��1� � C†�100�� 1�SC�000� � 0

(24)

▪ First order in �

C†�000�SC�010��1� � C†�010�� 1�SC�000� � 0

(25)▪ First order in �

C†�000�SC�001��0� � C†�010��0�SC�000�

� C†�000�S�001�C�000� � 0 (26)▪ of order ��

C†�000�SC�011��1� � C†�011�� 1�SC�000�

� C†�010�� 1�SC�001��0�

� C†�001��0�SC�010��1� � C†�010�� 1�S�001�C�000�

� C†�000�S�001�C�010��1� � 0 (27)

The perturbative expansion for the density ma-trix Eq. (12) leads to the following expression forthe density matrix at different orders

D�100��1� � C�000�nC†�100�� 1� � C�100��1�nC†�000�

D�010��1� � C�000�nC†�010�� 1� � C�010��1�nC†�000�

D�001��0� � C�000�nC†�001��0� � C�001��0�nC†�000�

(28)

D�011��1� � C�000�nC†�011�� 1� � C�011��1�nC†�000�

� C�010��1�nC†�001��0� � C�001��0�nC†�010�� 1� (29)

Now we use Eqs. (18)–(29) to derive an expressionfor ��(�) without explicitly determining themixed derivative of the density matrix (D(011)(1)).The derivation is accomplished in several steps. Inthe following we mention the main steps withoutgiving explicit results of each steps.

Step 1 Start with the positive frequency part of Eq.(22) and multiply from the left withC†(100)(�1).

Step 2 Take the adjoint of the positive frequencypart of Eq. (19) and multiply from the rightby C(011)(�1).

Step 3 Subtract the result of Step 2 from the resultof Step 1 and take the trace after multiplyingwith the occupation number matrix n.

Step 4 Take the adjoint of Eq. (22) at �� and mul-tiply from the right with C(100)(�1). Then,analogous to Step 2, multiply C†(011)(�1) to



the negative frequency part of Eq. (19) andrepeat Step 3.

Step 5 Next, add the equations resulting from Step3 and Step 4.

At this point we eliminate the second-order termsinvolving �(011) by using the first-order normaliza-tion condition Eq. (24) and express terms with C(011)

in terms of C(010) and C(001) by using the second-order normalization conditions.

Step 6 Add the negative of Tr[n(C†(001)(0)F(100)(�1)C(010)(�1) � C†(010)(�1)F(100)(�1)C(001)(0))]to both sides of the equation of the previousstep to obtain D(011) on the left-hand sideand using the following expressions

F�100��1� � Hext�100� � f̃HXCD�100��1�

F�011��1� � Hext�011� � f̃HXCD�011��1� � g̃HXCD�010��1�.

(30)

Here, the f̃HXC and g̃HXC matrices are given by

� f̃HXC�� drdr��r��r� fHXC�r, r�; ��r��r��

� g̃HXC�� drdr��r��r� fHXC�r, r�; ��L�� r��

� L�� r��r��r�� (31)

with

fHXC�r, r�; �� 1

�r � r�� fXC�r, r�; ��, (32)

where fXC(r, r�; �) is the exchange-correlation kernelfor which expressions based on the adiabatic local

density approximation (ALDA) [20] are generallyemployed. The matrix element of the external Ham-iltonian Hext

(011) is given by

�Hext�011�� dr��r�Hext

�010��r�L�� r��r� (33)

with

L�� r� � ��r

2 � �R� � � R� ��

. (34)

In deriving the expression for g̃HXC we have as-sumed that the induced density 0

(001)(r) is zero for astatic magnetic field. This is consistent with the factthat we have neglected the changes in the currentdensity due to the static magnetic field, and conse-quently the exchange-correlation potential is afunctional of the density only.

Step 7 Substitute the expression for Tr[Hext(100)

D(011)(�1)] obtained in the last step of Eq. (5)and using following defining relations

C�100��1� � C�000�U��1�

C�010��1� � C�000�U��1�

C�001��0� � C�000�U��0�

C†�000�S�001�C�000� � ��

��100��1� � ��1�

��010��1� � ��1�

��001��0� � ��0�

D�100��1� � D��1�

D�010��1� � D��1�

�Hext�011�� Hext

��

�Hext�101�� Hext

��, (35)

we get

��

�Hext��D��

��1� � ��

�Hext��D��

��1� � ��

��

�g̃HXC��D��1�D��

��1� � � Tr�n�U†��

� ��1��U��1� � U†��1��U��1�� Tr�n�U†��1��U��1� � U†��1��U��1��0��

� Tr�n�U†��0��1�U��1� � U†��1��1�U��0� � U†��1��1�U��0� � U†��0��

� ��1�U��1� � U†��1��0�U��1� � U†��1��0�U��1�� Tr�n�U†��1�U��1�

� U†��1�U��1��0�� Tr�n�U†��1�U†��0� � U��0�U��1��1��

� Tr�n�U†��1�U†��0� � U��0�U��1��1�� (36)


576 VOL. 101, NO. 5

In the above expansion the first two terms in-volve a sum over all atomic orbitals, and all theremaining terms need summation over all occupiedmolecular orbitals. Moreover, to obtain above ex-pression we make use of the fact that �†(�,�)(�1) ��(�,�)(�1) and �†(�,�)(�1) � �(�,�)(�1). The expres-sions for the U, �, �, and � matrices are given by

Upq��1� �

�pq��1�

�q�0� � �p

�0� � �

Upq��1� �

�pq��1�

�q�0� � �p

�0� � �

Upq��

�pq��0� � �q

�0��

�q�0� � �p

�0� , (37)

�pq��1� � C†�000�F�100��1�C�000�

�pq��1� � C†�000�F�010��1�C�000�

�pq��0� � � �

��

C*�p�000�C�q

�000��r��

2 � ��

�� r�

2 � �R� � � R� ��

H�000��, (38)

where H(000) is the unperturbed KS Hamiltonianand the vector r�� r� � R� � denotes the position ofthe electron relative to the atomic center �. We notethat the above equations are all valid for the occu-pied-virtual and the virtual-occupied blocks of thematrix only. For the occupied-occupied and thevirtual-virtual block we use [18, 21, 22]

Upq��,��1� � 0

Upq��0� � �1

2 ��. (39)

Accordingly the first-order � matrices are zerowhen p � occupied orbitals, q � virtual orbitals orvice versa. On the other hand, the occupied-occu-pied and the virtual-virtual blocks of the first-order� matrices are given by

�pq��,��1� � �pq

��,��1�

�pq��0� � �pq

��0� �12 �pq

��p�0� � �q

�0��. (40)

Finally, the �(�) and Hext(��) matrices read

�pq��

��

C*�p�000�C�q

�000�� r�2 � �R� � � R� ��

�

�� ,

(41)

and

�Hext�� dr��r�Hext

�100��r�L�� r��r�. (42)

The expression given by Eq. (36) constitutes themain result of this article. In next section we discussits similarity with the expression for the third-ordernonlinear optical hyperpolarizability and a schemeof numerical implementation of this expression.

III. Discussion and ConcludingRemarks

The above expression given in Eq. (36) alongwith the defining Eqs. (37)–(42) serve as workingrelations for computation of the Verdet constantwithin TDDFT. We note that in accordance with the(2n � 1)-rule of the perturbation theory, Eq. (36) iscompletely determined by the first-order perturbedquantities. Moreover, we point out that the expres-sion for the nonlinear optical hyperpolarizability��(�; 0, �) characterizing the electrooptic Pockeleffect (EOPE) is similar to the expression of Eq. (36),but the first five terms involving the change of theatomic orbitals and the overlap matrix �(�) are ab-sent [18, 24]. In the Amsterdam Density Functional(ADF) [23] package the computation of ��(�; 0, �)has already been implemented [24], and the matri-ces U(�,�)(1), �(�,�)(1), and �(�,�)(1) are avail-able. In the EOPE calculation Hext

(�)(0) corresponds toa static electric field. On the other hand, for thecalculation of the Verdet constant the static electricfield needs to be replaced by a static magnetic field.Therefore, the new ingredients required for compu-tation of the Verdet constant are �(�)(0), �(�)(0), and�(�) arising due to the application of static a mag-netic field. Implementation of a program for thecalculation of the Verdet constant based on Eq. (36)is currently in progress.

To conclude, this article is devoted to the deri-vation of an analytical expression for the Verdetconstant of MOR within TDDFT. For this purposewe employ the GIAOs, which results in an analyt-ical expression for the Verdet constant that is freefrom the gauge origin problem. The similarity of



this expression with the expression for the coeffi-cient of EOPE was also discussed. The results of ourimplementation of this expression in ADF packagewill be reported in the future.

References

1. Faraday, M. Philos Mag 1846, 28, 294.2. Faraday, M. Philos Trans R Soc (London) 1846, 136, 1.3. Verdet, E. M. Ann Chim, 3rd Ser 1854, 41, 370.4. Buckingham, A. D.; Stephens, P. J. Annu Rev Phys Chem

1966, 17, 399.5. Stephens, P. J. Annu Rev Phys Chem 1974, 25, 201.6. Stephens, P. J. Adv Chem Phys 1976, 35, 197.7. Chang, T. Y. J Chem Phys 1971, 54, 1433.8. Svendsen, E. N.; Hameka, H. F. Int J Quantum Chem 1974, 8,

789.9. Bishop, D. M.; Cybulski, S. M. J Chem Phys 1990, 93, 590.

10. Parkinson, W. A.; Sauer, S. P.; Oddershede, J.; Bishop, D. M.J Chem Phys 1993, 98, 487.

11. Jaszunski, M.; Jorgensen, P.; Rizzo, A.; Ruud, K.; Helgaker,T. Chem Phys Lett 1994, 222, 263.

12. Coriani, S.; Hattig, C.; Jorgensen, P.; Halkier, A.; Rizzo, A.Chem Phys Lett 1997, 281, 445.

13. Hansen, A.; Bouman, T. D. J Chem Phys 1985, 82, 5035.

14. Kutzelnigg, W. Isr J Chem 1980, 19, 193.

15. London, F. J Phys Radium 1937, 8, 397.

16. Ditchfield, R. Mol Phys 1974, 27, 789.

17. Schreckkenbach, G.; Ziegler, T. J Phys Chem 1995, 99, 606.

18. Karna, S. P.; Dupuis, M. J Comput Chem 1990, 12, 487.

19. Quinet, O.; Champagne, B. J Chem Phys 2001, 115, 6293.

20. Gross, E. K. U.; Dobson, J. F.; Petersilka, M. Top Curr Chem1996, 181, 81.

21. Handy, N. C.; Amos, R. D.; Gaw, J. F.; Rice, J. E.; Simandiras,E. D. Chem Phys Lett 1985, 120, 151.

22. Gonze, X. Phys Rev A 1995, 52, 1096.

23. Amsterdam Density Functional Program. http://www.scm.com.

24. van Gisbergen, S. Ph.D. Thesis; Virje Universiteit: Amster-dam, The Netherlands, 1999.


578 VOL. 101, NO. 5

A gauge-origin independent expression for the Verdet constant within the time-dependent density...

Documents

Transcript of A gauge-origin independent expression for the Verdet constant within the time-dependent density...