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Direct calculation of vertical ionization potentials by the coupled-cluster method
Volume-173,number 1
CHEMICALPHYSICSLETTERS
28 September 1990
Direct calculation of vertical ionization potentials by the coupled-cluster method Stanislav BiskupiE ‘, V. KvasniEka b and R. Klein a a Department ofPhysical Chemistry, Slovak Technical University,812 37 Bratisiava, Cxdwslovakia b Department OfMathematics Slovak Technical University,812 3 7 Bratislava, Czechoslovakia
Received 31 May 1990;in fina form 5 July 1990
The coupled-clustermethod in an open-shell version with non-Hen&an model interaction is applied for direct calculation of vertical ionization potentials of closed-shell molecular systems. The theory is illustrated by calculation of vertical ionization potentials of the water molecule. The calculated ionization potentials are in good agreement with experiment.
1. Iatroductlon
2. Theory
Much work has been done in the last two decades on the direct calculation of the vertical ionization potentials of atoms and molecules. Many authors have used the Green’s function technique to calculate corrections to Koopmans’ theorem (see, e.g., the excellent review article of Cederbaum and Domcke [ 11). Another efficient method based on the diagrammatic Rayleigh-Schriidinger perturbation theory has been developed and applied by Huba& Kvasnizka, and Urban [ 2-41. Recently, Kaldor [ 5-71 reported the application of an open-shell coupledcluster method (OSCCM) to the direct calculation of electron affinities, ionization potentials, and excitation energies. Kaldor’s treatment largely follows Lindgren’s normal-ordered formalism [ 81, The purpose of the present short communication is to apply a quasi-degenerate version of the coupledcluster method with non-Hermitian model interaction [ 91, based also on Lindgren’s normal-ordered formalism [ 8 1, to the direct calculation of ionization potentials of closed-shell molecular systems. In section 2 we outline the theoretical background of the method used whereas in section 3 we give an illustrative application to the water molecule studied within the DZ t P basis set of Gaussian atomic orbitals.
In this section we present basic ideas of our openshell version of the coupled-cluster method; more information and details can be found elsewhere [ 91. Let the model space Do be spanned by an unperturbed one-hole state vector, Do={I%)=&I%)}.
(1)
The exponential wave operator U=N[ e”] may, in this case, be significantly simplified as follows: V=iV[eT( 1 +Fh)] ,
(2)
where T is the correlation cluster operator [ 91 constructed with respect to I Qo) , and F,, corresponds to the relaxation cluster operator explicitly referred to the one-hole state 1CD,,). This second operator describes “relaxation” processes, when the hole state h is replaced by another hole state h’ and this is accompanied by simultaneous excitations. In this sense we can split the relaxation cluster operator Fh into two parts, where the first one corresponds to unexcited states and the second one to onefold (for 22 1) excited ones, r;, = Fh,o+ c Fh,l > 01
(3)
where 0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
33
Volume 173,number 1
CHEMICALPHYSICS LETTERS
WI
exponential wave operator U approximated as follows: U=
xziqx;
..x,:
x,’
4&,x,
,... X,,]
.
(4b)
For LO, the summation index h’ runs over all OG cupied hole states except that one indexed by h; this restriction comes from the requirement that the relaxation cluster operators Fh have only open components [91. By use of the non-Hermitian model eigenproblem [ 91, the vertical ionization potential (IP),, is determined by (IP)*=-el,td*.
28 Gptember 1990
1t T, + T, t Fh,l t Fh,2 ,
(8)
where T, and T2 are correlation cluster operators determined for the given closed-shell molecular system. The vertical ionization potential (IP)h was determined by (5 ), the diagrammatic terms contributing to Ahare displayed in fig. 1. Applying the general diagrammatic rules [ lo] for the algebraic interpretation of these diagrams, we get
(5)
The scalarentity Ah from the above equation is given by dh=(~*hlHIN[er(l+Fh)]I(Dh)U:,
(6)
where Z-Z,is the perturbation part of the total Hamiltonian and the subscript “IX” means that only linked and connected terms are taken into account. Now we can write a string of coupled nonlinear equations which determine the matrix elements of the relaxation cluster operator in the form appropriate for an actual implementation,
(hllh’)= A,+,’ A(h,h’) , h' -th
o~..d#,...hh =
(7a)
>
j--&y C Atpa,.a.Porrh, he,...ha.h’) ,
0)
LI
where DrY=~h,+...+~rhl-91-...-~pl- CA+ e,,. and the summation runs over all I! permutations l...l ( cyi...QI[ >. The matrix elements A(h, h’ ) and A(p,,...p,#, h,,...h,,h' ) are assigned to the matrix elements (@hh’IfflV@h)~,oaand
(@aI~~U@h)~.+wifha
fixed labelling of external lines [ 91. The subscript “op” means that only the so-called open diagrams are considered (cf. ref. [ 93). The general theory outlined above will be illustrated by its approximate version for closed-shellmolecular systems with the 34
m)
where (ZJ] v]KL),=Z(Z.Z~vlZU+)- (Z.Z)VILK). The many-body terms from the entity dh were separated into the so-called correlation and relaxation contributions, respectively.Perhaps such a partitioning of Ahmight be of value for deeper physical understanding and interpretation of physicalprocessesinvolved in the course of ionization of closed-shellmolecules, A careful analysisof these terms in the framework of second-order perturbation theory has been done by Cederbaum et al. [ 11.The matrix elements ( P1l]Z!Zi > and (PIP21(HJZ) in (9b) correspond to the correlation cluster operators T, and T,, respectively.They are determined by an independent coupled-cluster calculation of the ground-state correlation energy approxirnated by the first three terms in (8). The matrix elements corresponding to the relaxation cluster operators Fh,, and Fh,2are in general determined by
Volume 173,number1
28 September 1990
CHEMICAL PHYSICS LETTERS
h
h
h
h
h
h p2
PI
tf
4 Al
hl
4
A2
Bl
B2 h
h
h
h
hl Pl
D2
Cl
4
4
El Fig. 1. Diagrammatic representationof the
E2
coupled-cluster correctionsto the vertical ionizationpotential.
Furthermore, the matrix elements A ( , ) may be split into different terms originating in different diagrammatic terms [ 91.
3. Results and discussion The coupledcluster method in its approximate version outlined in section 2 has been applied to calculation of the first two vertical ionization potentials of the water molecule. The DZ+ P Gaussian basis set
[ll] (9s5pld contracted to 5s3pld with rl,=O.85 for oxygen and 5slp contracted to 3slp with qp= 1.0 for hydrogen) was used. The equilibrium geometry has heen used for all types of calculations (with bond lengthof 1.78127 auand bondangle 106.7”), i.e. the calculated ionization potentials are vertical. No orbitals are frozen in any of the correlation calculations. As a first step, we solved the coupled-cluster equations to obtain converged (i.e. self-consistent) values of matrix elements of TI and T2 (in the framework of the T, + T2 approximation, denoted by other authors [ 121 as the CCSD approximation). In the second step, these matrix elements are used in (9) for its iterative solution which yields the matrix elements of the relaxation operators F,,, and Fh,2. Then, finally, the matrix elements of both types are 35
Volume 173, number 1
28 September 1990
CHEMICAL PHYSICS LETTERS
Table 1 Vertical ionization potentials of the Hz0 molecule Level
Our results CCSD
lb, 3a,
12.60 14.77
Kaldor [7] CCSD+Q
12.67 14.84
Exp. [ 131
CCSD
OSCCM CCSD+T
12.16 14.44
12.73 14.92
OSCCM
12.62 (12.78) ‘) 14.74 (14.83) ”
‘) Values in parentheses are experimental ionization potentials corrected for zero-point vibration [ 141.
used for the calculation of correlation and relaxation contributions to the correction Ah. We did not use any numerical trick to accelerate the convergence of the iterative solution of nonlinear and coupled equations. The required accuracy (8% lo-* au) was achieved within eight iterations. Results obtained by us together with experimental data [ 13 ] and the results of Kaldor [ 7 ] are collected in table 1, Kaldor [ 7 ] has used the open-shell version of the coupled-cluster method for the direct calculation of excitation energies and ionization potentials. He demonstrated that the CCSD approximation does not yield satisfactory results and that inclusion of the triple excitation is highly desirable. Both these theoretical approaches (Kaldor’s OSCCM and our quasi-degenerate version of CCM) are, in fact, similar; they are different mainly in the manner of inclusion of correlation cluster operators in the process of construction of the relaxation cluster operator. Our calculations based on the Gaussian basis set comparable with that one used by Kaldor (he used a double-zeta plus polarization set of 6s4p 1d/3s 1p type) give slightly contradictory results. The CCSD approximation used in the framework of our approach gives relatively good results. To examine the influence of higher excitations, we have taken into account the 1T: terms in the calculation of matrix elements of the T2 operator (denoted as CCSD + Q in table 1). That is, in the iterative solution of coupled strings of equations for the matrix elements of Tz, we have included the part of tetra-excitations expressed by the operator f T $. One can see (table 1) that the inclusion of the f Tf term leads to significant improvement of the results. In conclusion, the present version of the coupled-
36
cluster method, applied to the direct calculation of vertical ionization potentials of the water molecule, yields these physical entities in fair agreement with experiment. In our forthcoming studies in this field, we will examine the influence of other correlation cluster operators on the resulting ionization potentials.
[ 1I L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. [ 21V. Kvasuitika and I. Huba& J. Chem. Phys. 60 ( 1974) 4483. [ 3 ] I. Hubti, V. KvasnZka and A Holubec, Chem. Phys. Letters 23 (1973) 381. [4] I. HubaE and M. Urban, Theoret. Chim. Acta 45 ( 1977) 185. [ 51A. Haque and U. Kaldor, Chem. Phys. Letters 117 ( 1985) 347. [6] U. Kaldor and A. Haque, Chem. Phys. Letters 128 ( 1986) 45. [ 71 U. Kaldor, J. Chem. Phys. 87 (1987) 467. [8]1. Lindgren, Intern. J. Quantum Chem. Symp. 12 (1978) 362. [ 91 V. KvasniEka, V. Laurinc, S. BiskupiE and M. Haring, Advan. Chem. Phys. 52 (1983) 182. [ 101 V. Kvasni&e, Advan. Chem. Phys. 36 ( 1977) 345. [ 111 S.Huzinaga, J. Chem. Phys. 42 (1965) 1293. [ 121G.D. Purvis and R.J. Bartlett, J. Chem. Phys. 76 ( 1982) 1919. [ 131A.W. Potts and WC. Price, Proc. Roy. Sot. A 326 (1972) 181. [14]W.Meyer,1ntern.J.QuantumChem.5(1971)341.
CHEMICALPHYSICSLETTERS
28 September 1990
Direct calculation of vertical ionization potentials by the coupled-cluster method Stanislav BiskupiE ‘, V. KvasniEka b and R. Klein a a Department ofPhysical Chemistry, Slovak Technical University,812 37 Bratisiava, Cxdwslovakia b Department OfMathematics Slovak Technical University,812 3 7 Bratislava, Czechoslovakia
Received 31 May 1990;in fina form 5 July 1990
The coupled-clustermethod in an open-shell version with non-Hen&an model interaction is applied for direct calculation of vertical ionization potentials of closed-shell molecular systems. The theory is illustrated by calculation of vertical ionization potentials of the water molecule. The calculated ionization potentials are in good agreement with experiment.
1. Iatroductlon
2. Theory
Much work has been done in the last two decades on the direct calculation of the vertical ionization potentials of atoms and molecules. Many authors have used the Green’s function technique to calculate corrections to Koopmans’ theorem (see, e.g., the excellent review article of Cederbaum and Domcke [ 11). Another efficient method based on the diagrammatic Rayleigh-Schriidinger perturbation theory has been developed and applied by Huba& Kvasnizka, and Urban [ 2-41. Recently, Kaldor [ 5-71 reported the application of an open-shell coupledcluster method (OSCCM) to the direct calculation of electron affinities, ionization potentials, and excitation energies. Kaldor’s treatment largely follows Lindgren’s normal-ordered formalism [ 81, The purpose of the present short communication is to apply a quasi-degenerate version of the coupledcluster method with non-Hermitian model interaction [ 91, based also on Lindgren’s normal-ordered formalism [ 8 1, to the direct calculation of ionization potentials of closed-shell molecular systems. In section 2 we outline the theoretical background of the method used whereas in section 3 we give an illustrative application to the water molecule studied within the DZ t P basis set of Gaussian atomic orbitals.
In this section we present basic ideas of our openshell version of the coupled-cluster method; more information and details can be found elsewhere [ 91. Let the model space Do be spanned by an unperturbed one-hole state vector, Do={I%)=&I%)}.
(1)
The exponential wave operator U=N[ e”] may, in this case, be significantly simplified as follows: V=iV[eT( 1 +Fh)] ,
(2)
where T is the correlation cluster operator [ 91 constructed with respect to I Qo) , and F,, corresponds to the relaxation cluster operator explicitly referred to the one-hole state 1CD,,). This second operator describes “relaxation” processes, when the hole state h is replaced by another hole state h’ and this is accompanied by simultaneous excitations. In this sense we can split the relaxation cluster operator Fh into two parts, where the first one corresponds to unexcited states and the second one to onefold (for 22 1) excited ones, r;, = Fh,o+ c Fh,l > 01
(3)
where 0009-2614/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
33
Volume 173,number 1
CHEMICALPHYSICS LETTERS
WI
exponential wave operator U approximated as follows: U=
xziqx;
..x,:
x,’
4&,x,
,... X,,]
.
(4b)
For LO, the summation index h’ runs over all OG cupied hole states except that one indexed by h; this restriction comes from the requirement that the relaxation cluster operators Fh have only open components [91. By use of the non-Hermitian model eigenproblem [ 91, the vertical ionization potential (IP),, is determined by (IP)*=-el,td*.
28 Gptember 1990
1t T, + T, t Fh,l t Fh,2 ,
(8)
where T, and T2 are correlation cluster operators determined for the given closed-shell molecular system. The vertical ionization potential (IP)h was determined by (5 ), the diagrammatic terms contributing to Ahare displayed in fig. 1. Applying the general diagrammatic rules [ lo] for the algebraic interpretation of these diagrams, we get
(5)
The scalarentity Ah from the above equation is given by dh=(~*hlHIN[er(l+Fh)]I(Dh)U:,
(6)
where Z-Z,is the perturbation part of the total Hamiltonian and the subscript “IX” means that only linked and connected terms are taken into account. Now we can write a string of coupled nonlinear equations which determine the matrix elements of the relaxation cluster operator in the form appropriate for an actual implementation,
(hllh’)= A,+,’ A(h,h’) , h' -th
o~..d#,...hh =
(7a)
>
j--&y C Atpa,.a.Porrh, he,...ha.h’) ,
0)
LI
where DrY=~h,+...+~rhl-91-...-~pl- CA+ e,,. and the summation runs over all I! permutations l...l ( cyi...QI[ >. The matrix elements A(h, h’ ) and A(p,,...p,#, h,,...h,,h' ) are assigned to the matrix elements (@hh’IfflV@h)~,oaand
(@aI~~U@h)~.+wifha
fixed labelling of external lines [ 91. The subscript “op” means that only the so-called open diagrams are considered (cf. ref. [ 93). The general theory outlined above will be illustrated by its approximate version for closed-shellmolecular systems with the 34
m)
where (ZJ] v]KL),=Z(Z.Z~vlZU+)- (Z.Z)VILK). The many-body terms from the entity dh were separated into the so-called correlation and relaxation contributions, respectively.Perhaps such a partitioning of Ahmight be of value for deeper physical understanding and interpretation of physicalprocessesinvolved in the course of ionization of closed-shellmolecules, A careful analysisof these terms in the framework of second-order perturbation theory has been done by Cederbaum et al. [ 11.The matrix elements ( P1l]Z!Zi > and (PIP21(HJZ) in (9b) correspond to the correlation cluster operators T, and T,, respectively.They are determined by an independent coupled-cluster calculation of the ground-state correlation energy approxirnated by the first three terms in (8). The matrix elements corresponding to the relaxation cluster operators Fh,, and Fh,2are in general determined by
Volume 173,number1
28 September 1990
CHEMICAL PHYSICS LETTERS
h
h
h
h
h
h p2
PI
tf
4 Al
hl
4
A2
Bl
B2 h
h
h
h
hl Pl
D2
Cl
4
4
El Fig. 1. Diagrammatic representationof the
E2
coupled-cluster correctionsto the vertical ionizationpotential.
Furthermore, the matrix elements A ( , ) may be split into different terms originating in different diagrammatic terms [ 91.
3. Results and discussion The coupledcluster method in its approximate version outlined in section 2 has been applied to calculation of the first two vertical ionization potentials of the water molecule. The DZ+ P Gaussian basis set
[ll] (9s5pld contracted to 5s3pld with rl,=O.85 for oxygen and 5slp contracted to 3slp with qp= 1.0 for hydrogen) was used. The equilibrium geometry has heen used for all types of calculations (with bond lengthof 1.78127 auand bondangle 106.7”), i.e. the calculated ionization potentials are vertical. No orbitals are frozen in any of the correlation calculations. As a first step, we solved the coupled-cluster equations to obtain converged (i.e. self-consistent) values of matrix elements of TI and T2 (in the framework of the T, + T2 approximation, denoted by other authors [ 121 as the CCSD approximation). In the second step, these matrix elements are used in (9) for its iterative solution which yields the matrix elements of the relaxation operators F,,, and Fh,2. Then, finally, the matrix elements of both types are 35
Volume 173, number 1
28 September 1990
CHEMICAL PHYSICS LETTERS
Table 1 Vertical ionization potentials of the Hz0 molecule Level
Our results CCSD
lb, 3a,
12.60 14.77
Kaldor [7] CCSD+Q
12.67 14.84
Exp. [ 131
CCSD
OSCCM CCSD+T
12.16 14.44
12.73 14.92
OSCCM
12.62 (12.78) ‘) 14.74 (14.83) ”
‘) Values in parentheses are experimental ionization potentials corrected for zero-point vibration [ 141.
used for the calculation of correlation and relaxation contributions to the correction Ah. We did not use any numerical trick to accelerate the convergence of the iterative solution of nonlinear and coupled equations. The required accuracy (8% lo-* au) was achieved within eight iterations. Results obtained by us together with experimental data [ 13 ] and the results of Kaldor [ 7 ] are collected in table 1, Kaldor [ 7 ] has used the open-shell version of the coupled-cluster method for the direct calculation of excitation energies and ionization potentials. He demonstrated that the CCSD approximation does not yield satisfactory results and that inclusion of the triple excitation is highly desirable. Both these theoretical approaches (Kaldor’s OSCCM and our quasi-degenerate version of CCM) are, in fact, similar; they are different mainly in the manner of inclusion of correlation cluster operators in the process of construction of the relaxation cluster operator. Our calculations based on the Gaussian basis set comparable with that one used by Kaldor (he used a double-zeta plus polarization set of 6s4p 1d/3s 1p type) give slightly contradictory results. The CCSD approximation used in the framework of our approach gives relatively good results. To examine the influence of higher excitations, we have taken into account the 1T: terms in the calculation of matrix elements of the T2 operator (denoted as CCSD + Q in table 1). That is, in the iterative solution of coupled strings of equations for the matrix elements of Tz, we have included the part of tetra-excitations expressed by the operator f T $. One can see (table 1) that the inclusion of the f Tf term leads to significant improvement of the results. In conclusion, the present version of the coupled-
36
cluster method, applied to the direct calculation of vertical ionization potentials of the water molecule, yields these physical entities in fair agreement with experiment. In our forthcoming studies in this field, we will examine the influence of other correlation cluster operators on the resulting ionization potentials.
[ 1I L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. [ 21V. Kvasuitika and I. Huba& J. Chem. Phys. 60 ( 1974) 4483. [ 3 ] I. Hubti, V. KvasnZka and A Holubec, Chem. Phys. Letters 23 (1973) 381. [4] I. HubaE and M. Urban, Theoret. Chim. Acta 45 ( 1977) 185. [ 51A. Haque and U. Kaldor, Chem. Phys. Letters 117 ( 1985) 347. [6] U. Kaldor and A. Haque, Chem. Phys. Letters 128 ( 1986) 45. [ 71 U. Kaldor, J. Chem. Phys. 87 (1987) 467. [8]1. Lindgren, Intern. J. Quantum Chem. Symp. 12 (1978) 362. [ 91 V. KvasniEka, V. Laurinc, S. BiskupiE and M. Haring, Advan. Chem. Phys. 52 (1983) 182. [ 101 V. Kvasni&e, Advan. Chem. Phys. 36 ( 1977) 345. [ 111 S.Huzinaga, J. Chem. Phys. 42 (1965) 1293. [ 121G.D. Purvis and R.J. Bartlett, J. Chem. Phys. 76 ( 1982) 1919. [ 131A.W. Potts and WC. Price, Proc. Roy. Sot. A 326 (1972) 181. [14]W.Meyer,1ntern.J.QuantumChem.5(1971)341.