This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 17915--17921 17915

Cite this: Phys. Chem.Chem.Phys.,2013,15, 17915

CAP/EOM-CCSD method for the study of potentialcurves of resonant states

Aryya Ghosh, Anagha Karne, Sourav Pal and Nayana Vaval*

The equation-of-motion coupled-cluster (EOM-CC) method along with the complex absorbing potential

(CAP) is used for the study of resonance in e�–N2 and e�–CO. Resonance position and width are studied

as a function of bond length. We report the potential curves (PC) of the resonance states.

I. Introduction

Resonant1 states are the quasi-bound or metastable states formedby the temporary trapping of an electron in an atom or amolecule. They occur in electron molecule scattering and provideinformation about the molecular system. Since these are quasi-bound they eventually decay into an atom or a molecule and a freeelectron. Generally, resonance2–5 scattering happens on a femto-second time scale. These metastable states6,7 are characterized bycomplex eigenvalues known as Siegert energies,8,9

Eres = ER � iG/2, (1)

where ER gives the position of the resonance and G is associatedwith the decay width. The lifetime can be obtained using therelation t = �h/G.

Since the resonance states are quasi-bound, the associatedwave function is not square integrable. It is desirable to convert theresonance problem into a bound state problem so that all thetechniques of the bound state problem can be used for the study ofthe resonance problem. Resonance states can also be describedas discrete states coupled with the continuum. The resonanceproblem is a challenging problem as we need to includecontinuum as well as electron correlation simultaneously.

The continuum problem of resonance states can be handledvia analytic continuation of the Hamiltonian. The analyticalcontinuation of the Hamiltonian can be achieved via complexscaling10,11 or using the complex absorbing potential (CAP).12–15

Another way is to implement the complex basis function16 methoddeveloped by Moiseyev and McCurdy.17–19 The complex scalingmethod is quite successful in the calculation of atomic resonancestates. However, it is difficult to implement the complex scalingmethod for molecules. The CAP20 method is numerically simple

and easier to implement in any ab initio electronic structuremethods. The CAP21 is a one-body operator and it can be applied incombination with any quantum chemistry method used for thestudy of bound states. The main idea behind the CAP approach isto introduce an absorbing boundary condition in the exteriorregion of the target molecule which keeps the target moleculeunperturbed and makes wave function square integrable.

The electron correlation and relaxation are important for theaccurate calculation of resonance energies and decay widths of themetastable anionic states. The accuracy of the complex absorbingpotential (CAP) approach in the calculation of electron scatteringresonance depends on the extent of the electron correlation inthe description of a particular resonance state. The CAP potentialhas already been applied for the study of resonance states usingdifferent electronic structure methods, e.g. configuration inter-action (CI) and the electron propagator theories and the coupled-cluster (CC) based methods. Sommerfeld et al.6 and Santraet al.22 have applied the CAP potential for the first time in thecontext of the CI method. Later on, Santra and co-workers23 haveapplied this method within the propagator theories. Pal andco-workers24,25 have implemented the CAP method within thecoupled-cluster (CC) framework. Recently, Sommerfeld andco-workers26 have implemented the CAP potential within thesymmetry-adapted cluster configuration interaction (SAC-CI)method. Among all these methods, the CC method27–29 givessignificantly improved results. The coupled-cluster method, due tothe presence of an exponential wave operator, includes dynamicelectron correlation in an efficient manner and maintains sizeextensivity. The EOM-CC method30–36 also provides a uniformtreatment of non-dynamic electron correlation in an efficientmanner. The calculation of direct energy difference is amenableusing the EOM-CC method. This makes the EOM-CC methodappropriate for the calculation of resonance energy.25

The potential curves of low energy resonance states haveimmense importance in different areas. They can be applied to

Physical Chemistry Division, CSIR-National Chemical Laboratory, Pune 411008,

India. E-mail: np.vaval@ncl.res.in

Received 19th June 2013,Accepted 30th August 2013

DOI: 10.1039/c3cp52552j

www.rsc.org/pccp

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17916 Phys. Chem. Chem. Phys., 2013, 15, 17915--17921 This journal is c the Owner Societies 2013

study the orientation of the absorbate molecules on crystal sur-faces. They can also be used as an alternative tool of extended X-rayabsorption fine structure (EXAFS) spectroscopy.37 In the presentpaper, we have calculated the potential curves for the low energy2Pg resonance state of N2

� and the 2P resonance state of CO�

employing a complex absorbing potential (CAP) in combinationwith the equation-of-motion coupled-cluster singles and doubles(EOM-CCSD) wave function. This method has been termed theCAP/EOM-CCSD method.

The paper is organized as follows. In Section II, we brieflydescribe the equation-of-motion coupled-cluster theory alongwith the CAP approach. In Section III, we discuss the potentialcurves of 2Pg N2

� and 2P CO�. Finally, we present our remarksin Section IV.

II. Theory

In this paper, we use CAP along with the EOM-CC method forthe calculation of resonance energies of e�–N2 and e�–CO fordifferent bond lengths. In this approach the outgoing electronis absorbed by CAP which is added to the physical Hamiltonian,making it a complex-symmetric, non-Hermitian Hamiltonian.

H(Z) = H � iZW, (2)

where Z is a real positive number representing the CAP strengthand W is the local semi-definite box type potential. This leads tothe Schrodinger equation

H(Z)c(Z) = E(Z)c(Z). (3)

In the presence of CAP, the original Hamiltonian now becomesnon-Hermitian. If one chooses an appropriate CAP, it evolves anabsorbing region where the Siegert eigenfunctions are damped. As aresult, the wave function becomes square integrable. However, theCAP introduces an artificial perturbation to the system and one canobtain the exact resonance eigenvalues and eigenfunctions in thelimit Z - 0 for a complete basis set. However, in practicalcomputation, one cannot solve the Siegert energy spectrum exactlybecause of the truncated basis set. The incomplete basis sets areused and one is forced to use the finite Z value. So, the onlypossibility is to minimize the CAP perturbation for a finite Z value.

To reduce the total error in the resonance energy calculationwe have used a stabilization method which is proposed by Jolicardand Austin12 and put on a firm ground by Riss and Meyer.38 Thereis an optimal Z value where the total error is minimum. In order toobtain the optimal Z value H(Z) is diagonalized for the number ofZ values. The combination of resulting eigenvalues gives an Ztrajectory. The Z trajectory is examined by using the logarithmicvelocity ui(Z). The resonance energy is explored through theminimization of logarithmic velocity

ui(Z) = ZqEi/qZ. (4)

The existence of a distinct minimum,

|ui(Zopt)| = min, (5)

gives the optimal Z value.

In our calculations we choose box-shaped CAP of the form

Wðx; cÞ ¼X3i¼1

Wi xi; cið Þ; (6)

where

Wi xi; cið Þ ¼0; xij j � ci;

xij j � cið Þ2; xij j4 ci

((7)

The ci (i = 1,2,3) represents the CAP box length in differentdirections. The xi (i = 1,2,3) is the bond length of the molecule indifferent coordinate axes from the center of the coordinate axis in acartesian coordinate system. This CAP can easily be represented in aGaussian basis set. It has been applied to the peripheral region ofthe target molecule to absorb the outgoing electron while keepingthe target unperturbed.

To obtain the resonance energies of anions CAP is combinedwith the EOM-CC method. Here we briefly review the EOM-CCmethod. In the EOM-CC approach, the coupled-cluster (CC)equation is solved first for the reference state. The exact groundstate wave function in the CC approach can be written as

|c0i = eT|f0i, (8)

where |f0i is the closed shell N-electron reference determinant, e.g.the restricted Hartree–Fock reference determinant (RHF), and T isthe cluster operator. The cluster operator T can be defined as followsin the coupled-cluster singles and doubles (CCSD) approximation

T ¼Xia

tai aaþai þ 1=4

Xab

Xij

tabij aaþabþaiaj (9)

where the standard convention for indices is used, i.e., indicesi, j refer to the occupied spin orbitals and indices a,b refer to theunoccupied spin orbitals. The cluster amplitudes t in eqn (9)are obtained by solving the coupled-cluster equations

hfai |(HN eT )c|f0i = 0, (10)

hfabij |(HN eT )c|f0i = 0, (11)

where fai and fab

ij are the excited determinants and the sub-script c in eqn (10) and (11) indicates that only connecteddiagrams are considered.

Within the EOM-CCSD formalism, the wave function for thekth electron attached state is obtained by the action of the Rk

operator on the ground state wave function |c0i

|cki = Rk|c0i. (12)

Rk is the linear excitation operator and it can be defined as

Rk ¼Xa

rakaþ þ 1=2

Xab

Xi

rabki aþbþi: (13)

The energy of the electron attached state is obtained by thefollowing equations:

e�THNRkeT|f0i = e�THN eTRk|f0i = EN+1Rk|f0i (14)

Defining�HN,�HN = e�THN eT � hf0|e�THN eT|f0i, (15)

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we can write�HNRk|f0i = wkRk|f0i. (16)

In matrix form eqn (16) can be written as�HNRk = wkRk, (17)

where wk is the energy change connected with the electron attach-ment process. So, in the EOM-CCSD method, the

�HN matrix is

constructed in the 1p and 2p1h space and diagonalized to obtainthe energies of electron attached states.

To obtain the resonance energy using EOM-CC in augmen-tation with CAP, CAP should be added at the single referencecoupled-cluster (SRCC) level to generate the complex T(Z)amplitudes, i.e.

|ck(Z)i = Rk(Z)|c0(Z)i (18)

where

|c0(Z)i = eT(Z)|f0i, (19)

which are then used to generate the�HN matrix.

�HN(Z) = e�T(Z)HN(Z)eT(Z) � hf0|e�T(Z)HN(Z)eT(Z)|f0i (20)

Diagonalization of this�HN(Z) gives the complex eigenvalues

wk(Z). However to obtain the resonance energies we need to usethe following equation since the ground state energies aresuppose to be CAP free:

Eres(Z) = wk(Z) + Esrcc(Z) � Esrcc(Z = 0) (21)

Thus, we lose the advantage of direct difference energy inthis approach. We call this approach CAP/EOM-CCSD-full. Also,in this procedure this entire step needs to be done for a fewhundred values of Z starting from Z = 0. The calculation ofSRCC for each Z value is time consuming and computationallydemanding since it scales as N6. The CAP is defined only overthe particle–particle block (virtual block) since it does not affectthe target system. The effect of CAP is very small on thecorrelation energy of the system. Hence, in the present paper,we have implemented an approximation for the inclusion ofCAP at the N + 1 electron wave function by keeping the groundstate |c0i CAP independent. We call this approach the CAP/EOM-CCSD method. Thus, we have

|ck(Z)i = Rk(Z)|c0i. (22)

We first solve the SRCC without any CAP potential. Thecluster amplitudes T(Z = 0) are scaled with the CAP potential byusing the following equations:

D1 = faa � fii, (23)

tai (Z) = [ta

i (Z = 0) � D1]/[D1 + Waa(Z)], (24)

D2 = faa + fbb � fii � fjj, (25)

tabij (Z) = [tab

ij (Z = 0) � D2]/[D2 + Waa(Z) + Wbb(Z)] (26)

With these new amplitudes T(Z) we construct the�HN matrix. We

have added CAP to the one-body particle–particle (�fpp) part of

the�HN matrix. So, the

�HN matrix can be written using eqn (20)

�HN(Z) = e�T(Z)HN(Z)eT(Z) � hf0|e�THNeT|f0i (27)

�HN(Z)Rk(Z) = wk(Z)Rk(Z) (28)

Thus the SRCC part is independent of the CAP perturbation,which means the N electron ground state is not affected by theCAP potential. The artificial nature of the CAP potential and itsapplication only to the particle–particle interaction part justifythe approximation of CAP/EOM-CCSD. The main advantage ofthis approximation is that it drastically reduces the Z trajectorygeneration time since SRCC calculation needs to be done onlyonce. Since the ground state is Z independent, resonanceenergy comes out to be the direct difference energy obtainedas eigenvalues of

�HN(Z) for different Z values. We have obtained

the complex eigenvalues after diagonalization of the�HN(Z)

matrix for different Z values starting from 0 to 0.01 with anincrement of 10�6. For the diagonalization purpose, we haveused the non-Hermitian version of the Davidson algorithm.39

By plotting the complex eigenvalues in the complex energyplane with the real part and the imaginary part of the energy asaxes, we get the Z trajectory. A resonance is identified with theappearance of minimum velocity.

uk = |Zqwk(Z)/qZ|. (29)

III. Results and discussion

In this paper we have studied the potential curves for the lowenergy metastable anionic resonance states using the CAP/EOM-CCSD method. We have investigated the potential curvesfor the 2Pg state of N2

� and the 2P state of CO�. The first stepin the CAP/EOM-CCSD computation is a SCF calculation for theneutral CO and N2 molecules. The SCF calculation has beenperformed by using the GAMESS40 suite of programs. The requiredmatrix elements of the EOM-CCSD and CAP matrices have beencomputed using our own codes.25

A. Potential curve for the 2Pg resonance state of N2�

In this section, we have calculated the potential curve (PC) forthe lower energy 2Pg state of N2

� using the CAP/EOM-CCSDmethod. In Fig. 1 we provide the PC for the 2Pg resonance stateof N2

�. The 2Pg state of N2� has been studied quite elaborately

both experimentally and theoretically and it is well establishedthat the ground state of N2

� is a 2Pg shape type resonance witha very short lifetime. The 2Pg state originates from the additionof an electron to the pg LUMO of the 1Sg

+ ground sate of theneutral N2 molecule and the associated electronic configura-tions of 1Sg

+ and 2Pg are

1Sg+: (core)4(1sg)2(1su)2(2sg)2(1pu)4 (30)

2Pg: (core)4(1sg)2(1su)2(2sg)2(1pu)4(npg)1. (31)

The calculated resonance energies (ER) and widths (G) forthe 2Pg state of N2

� are presented in Table 1 for differentinternuclear distances. The calculations have been carried outin four different basis sets. The first one consists of [5s7p]41

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contracted Gaussian type orbitals (CGTO). For each nitrogenatom, one p type (exponent = 0.03) function has been added. So,the basis set contains total [5s8p] functions on each nitrogenatom. We have denoted this basis as basis A. The second one isdenoted as basis B, the third one as basis C and the fourth one asbasis D. Basis B contains [5s10p]6 functions and basis C contains[5s10p2d]6 functions. Basis D consists of the d-aug-cc-pVDZ basisand two extra p type functions have been added. The exponent ofthe new p function is generated by scaling the exponent of thelast p function of the d-aug-cc-pVDZ basis by the ratio 1.5. Here,we have chosen the four different basis sets to show the basis setdependence on the PC of the 2Pg resonance state of N2

�.The CAP/EOM-CCSD computations are performed at a N–N

internuclear distance of 1.80–2.80 a.u. In CAP/EOM-CCSDcalculations, the two nitrogen atoms are placed in a cartesiancoordinate system at (0.0,0.0, �R/2), where R is the internucleardistance between two nitrogen atoms in the Z axis. The CAP boxside lengths are chosen to be cx = cy = dc and cz = dc + R/2. Inorder to obtain the optimum box size for which the absorbingpotential is best fitted to the basis set and the size of the targetsystem, the CAP/EOM-CCSD calculations were performed forthe various CAP box sizes. The velocity of the trajectory at thestabilization point is smallest for a box size dc = 2.50 a.u.compared to the other box size.

In Table 1 we report the results in basis A and basis Busing CAP/EOM-CCSD-full and CAP/EOM-CCSD methods at theequilibrium bond length. In basis A, the CAP/EOM-CCSD

method gives resonance position 1.98 eV and width 0.44 eV. It canbe seen that the CAP/EOM-CCSD method predicts the position ofresonance slightly lower (0.03 eV) compared to the CAP/EOM-CCSD-full method. However, the width is same in both the methods. Inbasis B, the CAP/EOM-CCSD method gives resonance position2.36 eV which is 0.02 eV lower compared to that from the CAP/EOM-CCSD-full method. Both the methods give identical resultsfor width in basis B. These results prove that the CAP/EOM-CCSDis a valid approximation for the study of resonance.

In Table 2 we report the position and width of resonance fordifferent bond lengths in four different basis sets using theCAP/EOM-CCSD method. It can be seen that in all four basissets the width of resonance is maximum at 1.80 a.u. and decreasesuniformly with increasing bond length and becomes zero at2.80 a.u. A similar trend is also observed for the position of theresonance. As we go from basis A to basis B, we get higherresonance position values for bond lengths up to 2.60 a.u. However,we get higher width values for bond lengths up to 2.07 a.u. Beyondthe equilibrium bond length the width of the resonance issmaller in basis B compared to basis A. This shows theimportance of addition of 2p functions. In basis C, which issimilar to basis B with just additional 2d functions, there is notmuch change in the position or width of resonance. This showsthat addition of d functions does not have much effect on theresonance position or width. In basis D we get results which aresimilar to those of basis B. It can be seen that in all the fourbasis sets qualitative trends are the same.

In Table 3 we report the position and width of resonance atthe equilibrium bond length using the CAP/EOMCCSD methodin basis B and C. We compare our results with other theoretical andexperimental results. The position and width of resonance usingthe linear algebraic method (2.13 eV, 0.31 eV), the Schwingermultichannel method (2.26 eV, 0.39 eV), and the R matrix(2.27 eV, 0.35 eV) are in good agreement with the experimentalvalues. The MRDCI (2.62 eV, 0.45 eV)47 method predicts aslightly higher value for the position of resonance. We alsoreport the TCAP/MR-CI results in basis B and C with differentactive spaces. In basis B, the CAP/EOM-CCSD method gives aresonance position of 2.36 eV with a width of 0.48 eV, whereasTCAP/MR-CI predicts a slightly higher position at 2.46 eV and a

Fig. 1 Potential curve (PC) for the 2Pg state of N2�. Diamonds indicate the PC in

basis B, triangles indicate the PC in basis A, squares indicate the PC in basis D, andcircles indicate the PC in basis C.

Table 1 Calculated resonance energies (ER) and decay widths (G) for the 2Pg

state of N2� using the CAP/EOM-CCSD-full method and the CAP/EOM-CCSD

method

Basis

CAP/EOM-CCSD-full CAP/EOM-CCSD

Energy (eV) Width (eV) Energy (eV) Width (eV)

Basis A 2.01 0.44 1.98 0.44Basis B 2.38 0.48 2.36 0.48

Table 2 Calculated resonance energies (ER) and decay widths (G) for the 2Pg

state of N2�

Internucleardistance (a.u.)

Basis A Basis B Basis C Basis D

ER

(eV)G(eV)

ER

(eV)G(eV)

ER

(eV)G(eV)

ER

(eV)G(eV)

1.80 1.98 0.78 2.36 0.84 2.36 0.87 2.43 0.811.90 1.93 0.65 2.25 0.68 2.34 0.71 2.45 0.652.00 1.98 0.60 2.28 0.57 2.26 0.59 2.36 0.392.07 1.98 0.44 2.36 0.48 2.28 0.48 2.36 0.352.20 1.92 0.39 2.06 0.29 2.01 0.32 1.98 0.292.30 1.74 0.30 1.82 0.27 1.74 0.21 1.60 0.182.40 1.33 0.17 1.46 0.16 1.41 0.08 1.33 0.112.50 1.27 0.04 1.38 0.11 1.36 0.05 1.05 0.092.60 1.25 0.02 1.27 0.08 1.30 0.05 0.97 0.052.70 1.25 0.01 0.97 0.02 1.30 0.02 0.95 0.042.80 1.25 0.00 0.95 0.00 1.30 0.00 0.93 0.00

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width of 0.41 eV. Inclusion of 2d functions reduces the position inCAP/EOM-CCSD (2.28 eV) and the TCAP/MR-CI6 predicts very smallvalues for position (1.69 eV) and width (0.20 eV) using 7 orbitals asan active space. On increasing the active space from 7 to 31 theTCAP/MR-CI method gives a resonance position of 2.86 eV and awidth of 0.64 eV. Inclusion of d functions does not affect the widthof resonance in CAP/EOM-CCSD; however, the position is slightlylowered. In the case of TCAP/MR-CI, for the 7 active space theposition and width are reduced drastically, but with the active spaceof 31 they are increased remarkably. In either case the TCAP/MR-CIresults are not in good agreement with the experimental values.

We also discuss the results obtained by Bartlett and co-workers48

for the shape resonance state of N2� using just bound state coupled-

cluster (CC) methods. They have reported the potential curve(PC) of N2

� and show its near co-incidence with the N2 groundstate for some range of R. However, we have found somediscrepancy in the potential curve (PC) of N2

� between theresults obtained from our calculations and the results obtainedby Bartlett and co-workers. In our calculations, we have shown thatthe 2Pg resonance state of N2

� is unstable by about 2.3 eV at theequilibrium bond length compared to the ground state energy ofN2 and it moves towards the bound state with increasing N–N bondlength. In our calculations, the PC of the 2Pg resonance state of N2

�

does not show co-incidence with the PC of the N2 ground state atnear equilibrium bond length. The energy difference between the2Pg resonance state of N2

� and the ground state N2 decreases whenthe N–N bond length increases.

B. Potential curve for the 2P state of CO�

In the last two decades, the low energy electron scattering of COmolecules has been of great interest to chemists. In particular,the 2P state of CO� is very interesting. The 2P state of CO� isquite similar to the 2Pg state of N2

� and consequently isexpected to be a short-lived shape type resonance. The 2P stateof CO� is generated by the addition of an electron to the pLUMO of the 1S+ ground state of a neutral CO molecule. Theassociated electronic configuration for the 2P state is

2P: (1s)2(2s)2(3s)2(4s)2(5s)2(1p)4(2p)1. (32)

In this section, we have calculated the potential curve (PC)for the 2P state of CO� using the CAP/EOM-CCSD method.

The PC for the 2P state of CO� are presented in Fig. 2. Two basissets are used to study the PCs of CO�. The first one consists of thed-aug-cc-pVDZ basis42 and the second one consists of the d-aug-cc-pVDZ+1P basis. The d-aug-cc-pVDZ+1P basis is constructed byadding one p function on carbon and oxygen atom each. Theexponent of the new p function is generated by scaling theexponent of the last p function in the d-aug-cc-pVDZ basis by afactor of 0.66. In the PC calculation, the CO molecule is placed ina cartesian coordinate system at (0.0,0.0, �R/2 a.u.) and the CAPbox side lengths are chosen to be cx = cy = dc and cz = dc + R/2,where R is the internuclear distance between the carbon and theoxygen atom. The optimum value of dc is 3.5 a.u. The calculatedresonance energies (ER) and decay widths (G) for the 2P resonancestate of CO� as a function of internuclear separation arepresented in Table 4.

From Table 4 it can be seen that in both the basis sets theposition and width G of resonance are reduced with the bondlength. The width of resonance approaches zero at 2.80 a.u. Aswe go from basis d-aug-cc-pVDZ to basis d-aug-cc-pVDZ+1P, inwhich we have an extra p function, the position and width ofresonance are reduced. From Fig. 2 it can be seen that PC inboth the basis sets have a similar trend.

We have also compared our results with the other theoreticaland experimental results available in the literature at the equili-brium bond length. The results are presented in Table 5. In thed-aug-cc-pVDZ basis, the EOM-CCSD method gives resonanceposition at 1.49 eV, which is very close to the experimental resultbut the width at 0.78 eV is overestimated compared to the

Table 3 Resonance energies (ER) and decay widths (G) for the 2Pg state of N2�

at the equilibrium bond length

Method Energy (eV) Width (eV)

Experimentala 2.32 0.41Linear algebraic methodb 2.13 0.31Schwinger multichannel methodc 2.26 0.39R matrixd 2.27/1.90 0.35/0.26MRDCI extrapolation methode 2.62 0.45CAP/CI f 2.97 0.65TCAP/MR-CIg (basis B) 2.46 0.41TCAP/MR-CIg (basis C) 2.86 0.64EOM-CCSD (basis B) 2.36 0.48EOM-CCSD (basis C) 2.28 0.48

a See ref. 43. b See ref. 44. c See ref. 45. d See ref. 46. e See ref. 47. f Seeref. 49. g See ref. 6.

Fig. 2 Potential curve (PC) for the 2P state of CO�. Squares indicate the PC in thed-aug-cc-pVDZ+1P basis and circles indicate the PC in the d-aug-cc-pVDZ basis.

Table 4 Calculated resonance energies (ER) and decay widths (G) for the 2Pstate of CO�

Internucleardistance (a.u.)

d-aug-cc-pVDZ d-aug-cc-pVDZ+1P

ER (eV) G (eV) ER (eV) G (eV)

1.80 1.41 1.09 1.38 0.681.90 1.36 0.97 1.32 0.602.00 1.46 0.92 1.33 0.562.13 1.49 0.78 1.30 0.522.20 1.49 0.54 1.36 0.482.30 1.38 0.48 0.98 0.362.40 1.25 0.24 0.89 0.302.50 0.87 0.21 0.72 0.122.60 0.70 0.08 0.57 0.102.70 0.68 0.05 0.35 0.052.80 0.70 0.00 0.35 0.00

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experimental value. When we add one p function on both thecarbon and oxygen atoms the resonance position is decreasedby 0.19 eV. In this basis, i.e., d-aug-cc-pVDZ+1P, we get theresonance position ER = 1.30 eV and width G = 0.52 eV, which isclose to the experimental value.

IV. Conclusion

In this paper, we have demonstrated the usefulness of the EOM-CCSD method for the study of metastable anionic resonance states.Here, we have applied this method to calculate the potential curvesfor the 2Pg state of N2

� and the 2P resonance state of CO�.From the potential curve study it can be seen that both the 2Pg

state of N2� and the 2P state of CO� follow a similar trend. Both

the decaying resonance states turn out to be a bound state at aninternuclear distance of 2.80 a.u.

Acknowledgements

The authors acknowledge the facilities of the Center of Excellence inScientific Computing at NCL. A. Ghosh acknowledges the Council ofScientific and Industrial Research (CSIR) for fellowship. S. Palacknowledges grant from DST, J. C. Bose Fellowship projecttowards completion of the work. N. Vaval and S. Pal acknow-ledge the CSIR MSM project for the financial support.

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Table 5 Calculated resonance energies (ER) and decay widths (G) for the 2Pstate of CO� at the equilibrium bond length

Method Energy (eV) Width (eV)

Experimenta 1.50 0.40Theoretical approaches:Boomerang modelb 1.52 0.80Close coupling methodc 1.75 0.28Second order dilated electronpropagator (real SCF)d

1.71 0.08

Results from the bi-orthogonaldilated electron propagator methode

Second order (S2) 1.68 0.09Diagonal 2p-h TDA (S2p-hTDA) 1.69 0.08Quasi-particle third order (Sq

3) 1.65 0.14Third order (S3) 1.65 0.14EOM-CCSD (d-aug-cc-pVDZ) 1.49 0.78EOM-CCSD (d-aug-cc-pVDZ+1P) 1.30 0.52

a See ref. 50. b See ref. 51. c See ref. 52. d See ref. 53. e See ref. 54.

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