- Email: [email protected]
Molecular property calculations for excited states using a multireference coupled-cluster approach
29 January 1999
Chemical Physics Letters 300 Ž1999. 125–130
Molecular property calculations for excited states using a multireference coupled-cluster approach Nayana Vaval, Sourav Pal
)
Physical Chemistry DiÕision, National Chemical Laboratory, Pune 411 008, India Received 27 July 1998; in final form 24 October 1998
Abstract In this Letter, we compute the properties of the ground-state and low-lying excited states of the water molecule dominated by quasi-degenerate single-hole–particle-excited determinants with respect to the restricted Hartree–Fock determinant of the ground state. We use an extensively correlated Fock-space-based multireference version of the coupled-cluster ŽCC. method. The dipole moments of the ground and low-lying vertically excited states of water at the equilibrium geometry have been calculated using the finite-field technique. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The single-reference coupled-cluster ŽSRCC. method w1,2x has been established as a powerful and compact technique for incorporating dynamical correlation in the calculation of electronic structure and spectra for non-degenerate systems. The SRCC method has been successfully used for the description of energy w2x, properties w3–5x and geometric derivatives w6x in the region around equilibrium. For properties and general derivatives, an analytic technique using the SRCC method has proved to be an efficient tool. Although several analytic stationary formulations have been recently proposed and tested w7–10x, the most standard technique approaching black-box character is based on a non-variational
) Corresponding author. Fax: q91 20 393044; E-mail: [email protected]
method w3x. For closed-shell systems, this has been well tested for electric as well as magnetic properties. The stationary analytic methods naturally incorporate a Ž2 n q 1.-type rule in evaluating the energy derivatives, simplifying the calculation of these. On the other hand, in the non-variational method, the first derivatives of T amplitudes with respect to the different components of the field can be eliminated in terms of a single set of additional de-excitation amplitudes for the computation of first-order properties w11x. The SRCC method with appropriate single-determinant model spaces can be used for limited open-shell cases, like high-spin systems, in particular. However, there are many open- or closed-shell cases away from equilibrium and excited states, in particular, where many determinants have nearly equal and dominant contributions to the wavefunction. In such cases, SRCC-based methods are still capable of addressing the problem in its generality. A more satisfactory solution is to start
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 3 1 - 1
126
N. VaÕal, S. Pal r Chemical Physics Letters 300 (1999) 125–130
from a reference model space consisting of dominant determinants and then incorporate electron correlation using the CC-based methods. This class of methods, known as multireference coupled-cluster ŽMRCC. methods, has been established as compact and conceptually correct to describe the problem. Extensive discussion of the MRCC theories can be found in the review by Mukherjee and Pal w12x. The general MRCC methods can be classified into two categories: Ža. Fock-space ŽFS. or valence universal w12–15x and Žb. Hilbert-space or state universal w12,16,17x. These are based on effective Hamiltonian-type theories where the eigenvalues of an effective Hamiltonian constructed within a smaller-dimension model space represent the corresponding exact state energies. The state universal methods, where every determinant acts as a vacuum, has been found to be more suitable for PES calculations w18,19x, particularly within an intermediate Hamiltonian w20x framework. On the other hand, the valence universal class, which is based on a common vacuum, has been shown to be useful for ionized w21,22x and low-lying excited states w23x. There are, however, few studies on the analytic derivative methods for the MRCC class of methods. Along the lines of Monkhorst w3x, a non-variational formulation of the valence universal MRCC method was developed by one of us w24x. A constrained variational method within the MRCC approach has been developed by Szalay w25x. However, these are at a preliminary stage of application. In this Letter, we use a finitefield-based valence universal MRCC method using a model space consisting of single-hole–particle-excited determinants to calculate the first-order properties of low-lying excited states of water. In this Letter, we present the finite-field calculations for electric field derivatives based on MRCC wavefunction. We have chosen for examples the low-lying excited states of H 2 O. We have chosen the ground-state-restricted Hartree–Fock ŽRHF. state of the H 2 O molecule as the vacuum, which defines the holes and particles. Active holes and particles are defined to be the ones contained in the model space determinants. The number of holes and particles contained in these determinants defines the rank of the Fock-space sector to which the model space belongs. For dipole moments of the low-lying excited states of H 2 O, we have chosen model space
consisting of one active hole and active particle. The dipole moment results have been compared with the numerical equation of motion-based CC ŽEOMCC. results available in the literature. In Section 2, we give a brief review of the valence universal MRCC. In Section 3, we discuss the computational details and in Section 4, we present the results for H 2 O and a discussion on these results.
2. Resume of the valence universal CC theory In this section, we will summarise the Fock-space multireference coupled-cluster ŽFSMRCC. approach used for the calculation of the excitation energies ŽEE.. In the FSMRCC approach, a convenient vacuum Žusually a RHF wavefunction of an N electron closed-shell system. is chosen with respect to which holes and particles are defined. The set of holes and particles is then divided into active and inactive sets. In general, any model space may consist of determinants with m active particles and n active holes. Such a model space is said to belong to an Ž m,n. sector, where the first index refers to the number of active particles and the second to the number of active holes in the model space. For a one-particle– one-hole N-electron state, the model space consists of the singly excited determinants composed of active particles and holes. The correlated wavefunction Cm corresponding to the model space Cm0 is generated by the action of the valence universal wave operator V on the model space. The valence universality implies that the wave operator has the flexibility to describe all the lower Fock-space sectors.
Ž 1.
i
where the Ci m are the model space coefficients for the model space F i with m active particles and n active holes. The correlated wavefunction in the MRCC is written as
Ž 2.
The valence universal V can be represented as
V s exp T Ž m , n. 4
Ž 3.
where the curly bracket denotes the normal ordering of operators enclosed in it. Since V must be able to
N. VaÕal, S. Pal r Chemical Physics Letters 300 (1999) 125–130
destroy any subset of m active particles and n active holes we have m
T Ž m , n. s
n
Ý Ý T Ž k ,l . .
Ž 4.
ks0 ls0
To calculate the energies and the amplitudes for the open-shell states, we solve the Bloch equation in Fock space and project it to the model space and the virtual space projector. The model space projection defines the effective Hamiltonian, the eigenvalues of which provide the energies of the system. The projection of the Bloch equation to the virtual space determinants yields the cluster amplitudes. A subsystem embedding condition ŽSEC. is prescribed to solve the equations of different valence sectors starting from the lowest sector upwards. In this process of SEC, the lower valence cluster amplitudes are frozen in the equations for higher valence amplitudes. The normal ordering of the ansatz ensures that in the solution of a particular sector equation, the amplitudes of higher sectors do not appear. Thus the SEC, along with normal ordering, effectively decouples the different valence sectors. The following equations are solved hierarchically upwards P Ž k ,l . w HV y V Heff x P Ž k ,l . s 0 ,
Ž 5.
Q Ž k ,l . w HV y V Heff x P Ž k ,l . s 0
Ž 6.
for k s 0,1, . . . m ;l s 0,1,2 . . . n . For the low-lying excited states we have used a model space consisting of determinants with one-particle–one-hole-excited determinants. Based on the RHF of the ground-state energies, we choose a set of active hole and particles and construct all singly excited determinants within this active set to build the model space. This is an incomplete model space. To make the model space complete we need to include, in addition to single h y p excited determinants, F HF , 2 p y 2 h, 3 p y 3h excited determinants and so on. The one-hole–one-particle model space, however, is a special case of incomplete model space, known as a quasi-complete model space. For this model space, the cluster operator can be decomposed as T Ž1,1. s T Ž0 ,0. q T Ž0 ,1. q T Ž1 ,0. q T Ž1 ,1.
Ž 7.
127
where T Ž0,0. is the cluster operator that acts on single determinant RHF. The operators T Ž0,1. and T Ž1,0. destroy exactly one active hole and one active particle, respectively. The T Ž1,1. operator, on the other hand, destroys both an active hole and an active particle. Each of these cluster operators can be decomposed into 1-, 2- and higher-body components. We have used the singles and doubles approximation for cluster operators at each Fock-space sector. The 1-body T1Ž1,1. operator is the de-excitation operator which takes model space to restricted Hartree–Fock determinant, which is outside the model space. The one-valence hole–particle problem is an incomplete model space and it has been shown that in such cases intermediate normalization is incompatible with the valence universality. Since valence universality is the key to the linked diagram theorem, intermediate normalization has to be abandoned in order for the linked diagram theorem w26x to be valid. This, in general, modifies the structure of the P space projection of the Bloch equation. Pal et al. w23x showed that for the special one-particle–one-hole model space there are simplifications which lead to the same P space projection equation as in the case of the complete model space obtained by the use of intermediate normalization. Further, the 1-body T1Ž1,1. operator does not contribute to the effective Hamiltonian and correspondingly to the energy of the excited state. Hence, although this operator is included in the wave operator, we have not considered its amplitudes. To evaluate the singlet and triplet excited states, one diagonalizes the following spin integrated form S T of matrices H EE and H EE , respectively: S Ž H EE . a p , b q s Ž HeffŽ0,1. . a b d p q q Ž HeffŽ1,0. . q p da b DŽ1 ,1. < y 2² a q < Heff pb : EŽ1 ,1. < q ² a q < Heff b p: ,
Ž 8.
T Ž H EE . a p , b q s Ž HeffŽ0,1. . a b d p q q Ž HeffŽ1,0. . q p da b EŽ1 ,1. < q ² a q < Heff b p: ,
Ž 9.
where a , b are active hole orbitals Žspatial. and p DŽ1,1. and q are active particle orbitals Žspatial.. Heff is the direct block of the Ž1,1. effective Hamiltonian EŽ1,1. and Heff is the corresponding exchange block.
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3. Computational details The FSMRCC method is used for the computation of low-lying vertical excitation energies and excited-state properties of the water molecule. As discussed in Section 2, we solve the equations using a SEC. We first solve the equation for the Ž0,0. sector, i.e the ground state. Then we construct an intermediate operator H ' Ž HeT Ž0,0. .conn . This term has closed and open diagrams. The closed part Ž Hcl . is the ground-state CC energy itself. If we drop this part in the equations for higher sectors, then we get the direct difference in energies at the ground-state geometry, i.e. the vertical difference energies. We truncate open parts of Ž H . to 1- and 2-body parts, denoted as F, V, respectively. Following the SEC, we solve Eqs. Ž5. and Ž6. for the Ž0,1. and Ž1,0. sectors and finally for the Ž1,1. sector. Thus in the process of calculating excitation energies ŽEE. we also get the ionization potential ŽIP. and electron affinity ŽEA.. This is the main advantage of the method used here over the other correlated methods. Also depending on the size of the model space, we get the energies of a multiple number of states from a single calculation. It also provides values of IP, EA, and EE which scale correctly with the size of the system. The energies of the vertically excited states are obtained by the addition of the closed part of H to the corresponding EE. The dipole moment of the vertically excited states are computed using the finite difference of the energies of the excited states at different field strengths. The method can also be used to obtain the adiabatic EE and properties at the equiblirium geometry of the excited states. For obtaining the adiabatic EE of a specific excited state, one needs to start from the equilibrium geometry of that particular state and then evaluate the energy of the state as outlined before. Finally, subtraction of the Ž H .cl at the ground-state geometry gives us the adiabatic EE of the particular state. For the dipole moment of the excited state at its equilibrium geometry, the energy of the excited state at this geometry has to be obtained at different field strengths. To obtain the adiabatic values for the excited state, a separate calculation has to be done for each excited state. However, in this Letter, we have reported only the vertical EE and the dipole moments of the vertically excited states of water.
The method, in singles and doubles approximation, scales roughly as N 6 , where N is the number of basis functions. Thus, this method can be used for medium sized molecules. For the water molecule, the calculations involving eight excited states were done on DEC Alpha 200 workstation and the computation of the energies of all the states took two days of computing time for each field.
4. Results and discussion In the study of the water molecule, we have used the polarized basis set of Sadlej w27x optimized for properties. The basis consists of a contracted set of 5s3p2d functions for the oxygen and 3s2p functions for the hydrogen atom. The equilibrium experimental geometry of the ground state of the water molecule has been chosen for this calculation. To save computational effort, we have frozen the lowest occupied and the two highest virtual orbitals during the calculations. This does not lead to any appreciable loss of accuracy in the results. We choose the RHF determinant of the ground state of the water molecule as a vacuum. The ground-state RHF wavefunction for water is
C HF s 1a21 2a211b 22 3a211b12
Ž 10 .
where the 1b 1 and 3a 1 are the highest and second highest occupied orbitals. The 3a 1 and 1b 1 are chosen as active holes and 3sa 1 and 3pb 2 are chosen as active particles. The low-lying excited states of water considered here are generated by single excitations from 3a 1 or 1b1 to 3sa 1 or 3pb 2 . The singlet and triplet EE thus generated have been reported in Table 1. The dominant excitations from the occupied to the virtual molecular orbitals have been indicated for each state. We have compared these with the experimental results, where available. Out of the eight excited states described here, the experimental EE for the 1 B 1Ž1b 1 ™ 3sa 1 . and 1A 1Ž3a 1 ™ 3sa 1 . states are well established. However, it may be noted that the experimental EE reported are the adiabatic EE, while the calculated values correspond to the vertical EE, which are larger than the corresponding adiabatic EE. The results presented in Table 1 show that the calculated values are larger than the corresponding experimental ones. The results are expected to be
N. VaÕal, S. Pal r Chemical Physics Letters 300 (1999) 125–130
129
Table 1 Excutation energies Žin eV. and dipole moment in Ža.u.. of H 2 O molecule at the equilibrium Ž R e . geometry obtained Sadlej basis set from MRCCSD calculations. Lowest occupied and highest virtual orbitals are frozen State
Excitation energy This work
Exp.
1
y 9.988 11.790 7.519 9.429 9.571 11.628 7.208 9.322
9.73 b – 7.49 b 9.1c 9.3 c – 7.0 d , 7.2 c 9.1d , 9.2 e , 8.9 c
A1 1 A 1Ž3a 1 ™ 3sa 1 . 1 B 2 Ž3a 1 ™ 3pb 2 . 1 B1Ž1b1 ™ 3sa 2 . 1 A 2 Ž1b1 ™ 3pb 2 . 3 A 1Ž3a 1 ™ 3sa 1 . 3 B 2 Ž3a 1 ™ 3pb 2 . 3 B1Ž1b1 ™ 3sa 1 . 3 A 2 Ž1b1 ™ 3pb 2 .
mCC SD
m EOMCC a
² m: a
0.728 y0.527 y0.622 y0.670 y0.452 y0.754 y0.609 y0.567 y0.437
0.724 y0.523 y0.603 y0.654 y0.539 y y y y
0.730 y0.485 y0.573 y0.632 y0.522 y y y y
a
Ref. w28x. Ref. w30x. c Ref. w31x. d Ref. w32x. e Ref. w33x. b
reliable due to the detailed treatment of both dynamical and non-dynamical correlation incorporated in the MRCC method. Earlier use of the FSMRCC method in the calculation of EE w23x also points to this. For the computation of dipole moments, we have obtained the energies of the excited state for three different field values applied along the C 2 axis. The fields chosen are of strength 0.005, 0.000 and y0.005 a.u. The derivatives have been obtained by a finite difference procedure. In addition, in the valence universal framework, the solution of the zero valence problem furnishes us the ground-state 1A 1 dipole moment value. Table 1 presents the dipole moments of these states. As a comparision, the dipole moment results of the singlet states obtained by the EOMCCSD approximation are presented using expectation value and numerical finite field techniques w28x. Although the ground-state of H 2 O has the largest dipole moment, there is no direct relation between the dipole moment values and the energies of the states. However, we observe a change of orientation of the dipole moments of the excited states with respect to the ground electronic state. As pointed out by Urban and Sadlej w29x, the change of direction of the dipole moment axis for 1 B 1 and 3 B1 arises predominantally from the change of the oxygen lone pair 1b 1 contribution. The virtual orbital 3sa 1 , which is dominantly occupied in these two states, gives a large negative contribution to the
dipole moment. However, we see a similar effect for dominant excitation from 1b 1 to 3pb 2 Ž3A 2 and 1A 2 states. as well as for 3A 1 , 1A 1 , 3 B 2 and 1 B 2 states dominated by excitation from 3a 1 to 3sa 1 or 3pb 2 . An extensive treatment of dynamical and non-dynamical correlation in MRCCSD method is expected to provide accurate values of the dipole moment. Comparing the MRCCSD results and EOMCCSD finite field and expectation value results for the singlet states, we observe that the results agree in general terms. However, the trend of the EOMCCSD and MRCCSD results is not the same. However, the trend of the two versions of the EOM CCSD results is the same, with the excited 1A 1 dipole moment values being the lowest. The EOM CCSD dipole moments calculated with the expectation value methods are lower than the finite field ones. Turning to the MRCC results, we observe that the values of the dipole moment are the smallest for the lowest A 2 state in both the singlet and triplet categories
Acknowledgements NV acknowledges the finantial support from Council of Scientific and Industrail Research ŽCSIR.. The authors acknowledge the Department of Science and Technology for funding. Dec Alpha 200 workstation was used for the computation of this work.
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References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x
J. Cizek, J. Chem. Phys. 45 Ž1966. 4156. R.J. Bartlett, Ann. Rev. Phys. Chem. 32 Ž1981. 359. H.J. Monkhorst, Int. J. Quantum Chem. S 11 Ž1977. 421. J. Noga, M. Urban, Theor. Chim. Acta 73 Ž1988. 291. H. Sekino, R.J. Bartlett, Int. J. Quantum Chem. S 18 Ž1984. 255. H. Koch, P. Jorgensen, J. Chem. Phys. 93 Ž1990. 3333. K.B. Ghose, P. Nair, S. Pal, Chem. Phys. Lett. 211 Ž1993. 15. N. Vaval, S. Pal, Phys. Rev. A 54 Ž1996. 250. S. Pal, Theor. Chim. Acta 66 Ž1984. 151. S. Pal, Phys. Rev. A 33 Ž1986. 2240. E.A. Salter, G.W. Trucks, R.J. Bartlett, J. Chem. Phys. 90 Ž1989. 1752. D. Mukherjee, S. Pal, Adv. Quantum Chem. 20 Ž1989. 291. W. Kutzelnigg, J. Chem. Phys. 77 Ž1982. 2081. I. Lindgren, Phys. Scr. 32 Ž1985. 291. D. Mukherjee, R.K. Moitra, A. Mukhopadhyay, Mol. Phys. 30 Ž1975. 1861. B. Jeziorski, H. Monkhorst, Phys. Rev. A 24 Ž1981. 1668.
w17x B. Jeziorski, J. Paldus, J. Chem. Phys. 8 Ž1988. 5673. w18x K.B. Ghose, S. Pal, J. Chem. Phys. 97 Ž1992. 3863. w19x J. Paldus, P. Piecuch, L. Pylypow, B. Jeziorski, Phys. Rev. A 47 Ž1993. 2738. w20x P. Durand, J.P. Malrieu, Adv. Chem. Phys. 67 Ž1987. 321. w21x S. Pal, M. Rittby, R.J. Bartlett, D. Sinha, D. Mukherjee, Chem. Phys. Lett. 137 Ž1987. 273. w22x N. Vaval, K.B. Ghose, S. Pal, D. Mukherjee, Chem. Phys. Lett. 209 Ž1993. 291. w23x S. Pal, M. Rittby, R.J. Bartlett, D. Sinha, D. Mukherjee, J. Chem. Phys. 88 Ž1988. 4357. w24x S. Pal, Int. J. Quantum Chem. 41 Ž1992. 443. w25x P. Szalay, Int. J. Quantum Chem. 55 Ž1995. 151. w26x D. Mukherjee, Chem. Phys. Lett. 125 Ž1986. 207. w27x A. Sadlej, Theor. Chim. Acta 79 Ž1991. 123. w28x J.F. Stanton, R.J. Bartlett, J. Chem. Phys. 98 Ž1993. 7029. w29x M. Urban, A.J. Sadlej, Theor. Chem. Acta 78 Ž1990. 189. w30x J.W.C. Johns, Can. J. Phys. 41 Ž1963. 209. w31x A. Chutjian, S. Trajmer, R.I. Hall, J. Chem. Phys. 63 Ž1975. 892. w32x F.W.E. Knoop, H.H. Brongersma, C.J. Oosterhoff, Chem. Phys. Lett. 13 Ž1972. 20. w33x G. Schulz, J. Chem. Phys. 33 Ž1960. 1661.
Chemical Physics Letters 300 Ž1999. 125–130
Molecular property calculations for excited states using a multireference coupled-cluster approach Nayana Vaval, Sourav Pal
)
Physical Chemistry DiÕision, National Chemical Laboratory, Pune 411 008, India Received 27 July 1998; in final form 24 October 1998
Abstract In this Letter, we compute the properties of the ground-state and low-lying excited states of the water molecule dominated by quasi-degenerate single-hole–particle-excited determinants with respect to the restricted Hartree–Fock determinant of the ground state. We use an extensively correlated Fock-space-based multireference version of the coupled-cluster ŽCC. method. The dipole moments of the ground and low-lying vertically excited states of water at the equilibrium geometry have been calculated using the finite-field technique. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The single-reference coupled-cluster ŽSRCC. method w1,2x has been established as a powerful and compact technique for incorporating dynamical correlation in the calculation of electronic structure and spectra for non-degenerate systems. The SRCC method has been successfully used for the description of energy w2x, properties w3–5x and geometric derivatives w6x in the region around equilibrium. For properties and general derivatives, an analytic technique using the SRCC method has proved to be an efficient tool. Although several analytic stationary formulations have been recently proposed and tested w7–10x, the most standard technique approaching black-box character is based on a non-variational
) Corresponding author. Fax: q91 20 393044; E-mail: [email protected]
method w3x. For closed-shell systems, this has been well tested for electric as well as magnetic properties. The stationary analytic methods naturally incorporate a Ž2 n q 1.-type rule in evaluating the energy derivatives, simplifying the calculation of these. On the other hand, in the non-variational method, the first derivatives of T amplitudes with respect to the different components of the field can be eliminated in terms of a single set of additional de-excitation amplitudes for the computation of first-order properties w11x. The SRCC method with appropriate single-determinant model spaces can be used for limited open-shell cases, like high-spin systems, in particular. However, there are many open- or closed-shell cases away from equilibrium and excited states, in particular, where many determinants have nearly equal and dominant contributions to the wavefunction. In such cases, SRCC-based methods are still capable of addressing the problem in its generality. A more satisfactory solution is to start
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 3 1 - 1
126
N. VaÕal, S. Pal r Chemical Physics Letters 300 (1999) 125–130
from a reference model space consisting of dominant determinants and then incorporate electron correlation using the CC-based methods. This class of methods, known as multireference coupled-cluster ŽMRCC. methods, has been established as compact and conceptually correct to describe the problem. Extensive discussion of the MRCC theories can be found in the review by Mukherjee and Pal w12x. The general MRCC methods can be classified into two categories: Ža. Fock-space ŽFS. or valence universal w12–15x and Žb. Hilbert-space or state universal w12,16,17x. These are based on effective Hamiltonian-type theories where the eigenvalues of an effective Hamiltonian constructed within a smaller-dimension model space represent the corresponding exact state energies. The state universal methods, where every determinant acts as a vacuum, has been found to be more suitable for PES calculations w18,19x, particularly within an intermediate Hamiltonian w20x framework. On the other hand, the valence universal class, which is based on a common vacuum, has been shown to be useful for ionized w21,22x and low-lying excited states w23x. There are, however, few studies on the analytic derivative methods for the MRCC class of methods. Along the lines of Monkhorst w3x, a non-variational formulation of the valence universal MRCC method was developed by one of us w24x. A constrained variational method within the MRCC approach has been developed by Szalay w25x. However, these are at a preliminary stage of application. In this Letter, we use a finitefield-based valence universal MRCC method using a model space consisting of single-hole–particle-excited determinants to calculate the first-order properties of low-lying excited states of water. In this Letter, we present the finite-field calculations for electric field derivatives based on MRCC wavefunction. We have chosen for examples the low-lying excited states of H 2 O. We have chosen the ground-state-restricted Hartree–Fock ŽRHF. state of the H 2 O molecule as the vacuum, which defines the holes and particles. Active holes and particles are defined to be the ones contained in the model space determinants. The number of holes and particles contained in these determinants defines the rank of the Fock-space sector to which the model space belongs. For dipole moments of the low-lying excited states of H 2 O, we have chosen model space
consisting of one active hole and active particle. The dipole moment results have been compared with the numerical equation of motion-based CC ŽEOMCC. results available in the literature. In Section 2, we give a brief review of the valence universal MRCC. In Section 3, we discuss the computational details and in Section 4, we present the results for H 2 O and a discussion on these results.
2. Resume of the valence universal CC theory In this section, we will summarise the Fock-space multireference coupled-cluster ŽFSMRCC. approach used for the calculation of the excitation energies ŽEE.. In the FSMRCC approach, a convenient vacuum Žusually a RHF wavefunction of an N electron closed-shell system. is chosen with respect to which holes and particles are defined. The set of holes and particles is then divided into active and inactive sets. In general, any model space may consist of determinants with m active particles and n active holes. Such a model space is said to belong to an Ž m,n. sector, where the first index refers to the number of active particles and the second to the number of active holes in the model space. For a one-particle– one-hole N-electron state, the model space consists of the singly excited determinants composed of active particles and holes. The correlated wavefunction Cm corresponding to the model space Cm0 is generated by the action of the valence universal wave operator V on the model space. The valence universality implies that the wave operator has the flexibility to describe all the lower Fock-space sectors.
Ž 1.
i
where the Ci m are the model space coefficients for the model space F i with m active particles and n active holes. The correlated wavefunction in the MRCC is written as
Ž 2.
The valence universal V can be represented as
V s exp T Ž m , n. 4
Ž 3.
where the curly bracket denotes the normal ordering of operators enclosed in it. Since V must be able to
N. VaÕal, S. Pal r Chemical Physics Letters 300 (1999) 125–130
destroy any subset of m active particles and n active holes we have m
T Ž m , n. s
n
Ý Ý T Ž k ,l . .
Ž 4.
ks0 ls0
To calculate the energies and the amplitudes for the open-shell states, we solve the Bloch equation in Fock space and project it to the model space and the virtual space projector. The model space projection defines the effective Hamiltonian, the eigenvalues of which provide the energies of the system. The projection of the Bloch equation to the virtual space determinants yields the cluster amplitudes. A subsystem embedding condition ŽSEC. is prescribed to solve the equations of different valence sectors starting from the lowest sector upwards. In this process of SEC, the lower valence cluster amplitudes are frozen in the equations for higher valence amplitudes. The normal ordering of the ansatz ensures that in the solution of a particular sector equation, the amplitudes of higher sectors do not appear. Thus the SEC, along with normal ordering, effectively decouples the different valence sectors. The following equations are solved hierarchically upwards P Ž k ,l . w HV y V Heff x P Ž k ,l . s 0 ,
Ž 5.
Q Ž k ,l . w HV y V Heff x P Ž k ,l . s 0
Ž 6.
for k s 0,1, . . . m ;l s 0,1,2 . . . n . For the low-lying excited states we have used a model space consisting of determinants with one-particle–one-hole-excited determinants. Based on the RHF of the ground-state energies, we choose a set of active hole and particles and construct all singly excited determinants within this active set to build the model space. This is an incomplete model space. To make the model space complete we need to include, in addition to single h y p excited determinants, F HF , 2 p y 2 h, 3 p y 3h excited determinants and so on. The one-hole–one-particle model space, however, is a special case of incomplete model space, known as a quasi-complete model space. For this model space, the cluster operator can be decomposed as T Ž1,1. s T Ž0 ,0. q T Ž0 ,1. q T Ž1 ,0. q T Ž1 ,1.
Ž 7.
127
where T Ž0,0. is the cluster operator that acts on single determinant RHF. The operators T Ž0,1. and T Ž1,0. destroy exactly one active hole and one active particle, respectively. The T Ž1,1. operator, on the other hand, destroys both an active hole and an active particle. Each of these cluster operators can be decomposed into 1-, 2- and higher-body components. We have used the singles and doubles approximation for cluster operators at each Fock-space sector. The 1-body T1Ž1,1. operator is the de-excitation operator which takes model space to restricted Hartree–Fock determinant, which is outside the model space. The one-valence hole–particle problem is an incomplete model space and it has been shown that in such cases intermediate normalization is incompatible with the valence universality. Since valence universality is the key to the linked diagram theorem, intermediate normalization has to be abandoned in order for the linked diagram theorem w26x to be valid. This, in general, modifies the structure of the P space projection of the Bloch equation. Pal et al. w23x showed that for the special one-particle–one-hole model space there are simplifications which lead to the same P space projection equation as in the case of the complete model space obtained by the use of intermediate normalization. Further, the 1-body T1Ž1,1. operator does not contribute to the effective Hamiltonian and correspondingly to the energy of the excited state. Hence, although this operator is included in the wave operator, we have not considered its amplitudes. To evaluate the singlet and triplet excited states, one diagonalizes the following spin integrated form S T of matrices H EE and H EE , respectively: S Ž H EE . a p , b q s Ž HeffŽ0,1. . a b d p q q Ž HeffŽ1,0. . q p da b DŽ1 ,1. < y 2² a q < Heff pb : EŽ1 ,1. < q ² a q < Heff b p: ,
Ž 8.
T Ž H EE . a p , b q s Ž HeffŽ0,1. . a b d p q q Ž HeffŽ1,0. . q p da b EŽ1 ,1. < q ² a q < Heff b p: ,
Ž 9.
where a , b are active hole orbitals Žspatial. and p DŽ1,1. and q are active particle orbitals Žspatial.. Heff is the direct block of the Ž1,1. effective Hamiltonian EŽ1,1. and Heff is the corresponding exchange block.
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3. Computational details The FSMRCC method is used for the computation of low-lying vertical excitation energies and excited-state properties of the water molecule. As discussed in Section 2, we solve the equations using a SEC. We first solve the equation for the Ž0,0. sector, i.e the ground state. Then we construct an intermediate operator H ' Ž HeT Ž0,0. .conn . This term has closed and open diagrams. The closed part Ž Hcl . is the ground-state CC energy itself. If we drop this part in the equations for higher sectors, then we get the direct difference in energies at the ground-state geometry, i.e. the vertical difference energies. We truncate open parts of Ž H . to 1- and 2-body parts, denoted as F, V, respectively. Following the SEC, we solve Eqs. Ž5. and Ž6. for the Ž0,1. and Ž1,0. sectors and finally for the Ž1,1. sector. Thus in the process of calculating excitation energies ŽEE. we also get the ionization potential ŽIP. and electron affinity ŽEA.. This is the main advantage of the method used here over the other correlated methods. Also depending on the size of the model space, we get the energies of a multiple number of states from a single calculation. It also provides values of IP, EA, and EE which scale correctly with the size of the system. The energies of the vertically excited states are obtained by the addition of the closed part of H to the corresponding EE. The dipole moment of the vertically excited states are computed using the finite difference of the energies of the excited states at different field strengths. The method can also be used to obtain the adiabatic EE and properties at the equiblirium geometry of the excited states. For obtaining the adiabatic EE of a specific excited state, one needs to start from the equilibrium geometry of that particular state and then evaluate the energy of the state as outlined before. Finally, subtraction of the Ž H .cl at the ground-state geometry gives us the adiabatic EE of the particular state. For the dipole moment of the excited state at its equilibrium geometry, the energy of the excited state at this geometry has to be obtained at different field strengths. To obtain the adiabatic values for the excited state, a separate calculation has to be done for each excited state. However, in this Letter, we have reported only the vertical EE and the dipole moments of the vertically excited states of water.
The method, in singles and doubles approximation, scales roughly as N 6 , where N is the number of basis functions. Thus, this method can be used for medium sized molecules. For the water molecule, the calculations involving eight excited states were done on DEC Alpha 200 workstation and the computation of the energies of all the states took two days of computing time for each field.
4. Results and discussion In the study of the water molecule, we have used the polarized basis set of Sadlej w27x optimized for properties. The basis consists of a contracted set of 5s3p2d functions for the oxygen and 3s2p functions for the hydrogen atom. The equilibrium experimental geometry of the ground state of the water molecule has been chosen for this calculation. To save computational effort, we have frozen the lowest occupied and the two highest virtual orbitals during the calculations. This does not lead to any appreciable loss of accuracy in the results. We choose the RHF determinant of the ground state of the water molecule as a vacuum. The ground-state RHF wavefunction for water is
C HF s 1a21 2a211b 22 3a211b12
Ž 10 .
where the 1b 1 and 3a 1 are the highest and second highest occupied orbitals. The 3a 1 and 1b 1 are chosen as active holes and 3sa 1 and 3pb 2 are chosen as active particles. The low-lying excited states of water considered here are generated by single excitations from 3a 1 or 1b1 to 3sa 1 or 3pb 2 . The singlet and triplet EE thus generated have been reported in Table 1. The dominant excitations from the occupied to the virtual molecular orbitals have been indicated for each state. We have compared these with the experimental results, where available. Out of the eight excited states described here, the experimental EE for the 1 B 1Ž1b 1 ™ 3sa 1 . and 1A 1Ž3a 1 ™ 3sa 1 . states are well established. However, it may be noted that the experimental EE reported are the adiabatic EE, while the calculated values correspond to the vertical EE, which are larger than the corresponding adiabatic EE. The results presented in Table 1 show that the calculated values are larger than the corresponding experimental ones. The results are expected to be
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Table 1 Excutation energies Žin eV. and dipole moment in Ža.u.. of H 2 O molecule at the equilibrium Ž R e . geometry obtained Sadlej basis set from MRCCSD calculations. Lowest occupied and highest virtual orbitals are frozen State
Excitation energy This work
Exp.
1
y 9.988 11.790 7.519 9.429 9.571 11.628 7.208 9.322
9.73 b – 7.49 b 9.1c 9.3 c – 7.0 d , 7.2 c 9.1d , 9.2 e , 8.9 c
A1 1 A 1Ž3a 1 ™ 3sa 1 . 1 B 2 Ž3a 1 ™ 3pb 2 . 1 B1Ž1b1 ™ 3sa 2 . 1 A 2 Ž1b1 ™ 3pb 2 . 3 A 1Ž3a 1 ™ 3sa 1 . 3 B 2 Ž3a 1 ™ 3pb 2 . 3 B1Ž1b1 ™ 3sa 1 . 3 A 2 Ž1b1 ™ 3pb 2 .
mCC SD
m EOMCC a
² m: a
0.728 y0.527 y0.622 y0.670 y0.452 y0.754 y0.609 y0.567 y0.437
0.724 y0.523 y0.603 y0.654 y0.539 y y y y
0.730 y0.485 y0.573 y0.632 y0.522 y y y y
a
Ref. w28x. Ref. w30x. c Ref. w31x. d Ref. w32x. e Ref. w33x. b
reliable due to the detailed treatment of both dynamical and non-dynamical correlation incorporated in the MRCC method. Earlier use of the FSMRCC method in the calculation of EE w23x also points to this. For the computation of dipole moments, we have obtained the energies of the excited state for three different field values applied along the C 2 axis. The fields chosen are of strength 0.005, 0.000 and y0.005 a.u. The derivatives have been obtained by a finite difference procedure. In addition, in the valence universal framework, the solution of the zero valence problem furnishes us the ground-state 1A 1 dipole moment value. Table 1 presents the dipole moments of these states. As a comparision, the dipole moment results of the singlet states obtained by the EOMCCSD approximation are presented using expectation value and numerical finite field techniques w28x. Although the ground-state of H 2 O has the largest dipole moment, there is no direct relation between the dipole moment values and the energies of the states. However, we observe a change of orientation of the dipole moments of the excited states with respect to the ground electronic state. As pointed out by Urban and Sadlej w29x, the change of direction of the dipole moment axis for 1 B 1 and 3 B1 arises predominantally from the change of the oxygen lone pair 1b 1 contribution. The virtual orbital 3sa 1 , which is dominantly occupied in these two states, gives a large negative contribution to the
dipole moment. However, we see a similar effect for dominant excitation from 1b 1 to 3pb 2 Ž3A 2 and 1A 2 states. as well as for 3A 1 , 1A 1 , 3 B 2 and 1 B 2 states dominated by excitation from 3a 1 to 3sa 1 or 3pb 2 . An extensive treatment of dynamical and non-dynamical correlation in MRCCSD method is expected to provide accurate values of the dipole moment. Comparing the MRCCSD results and EOMCCSD finite field and expectation value results for the singlet states, we observe that the results agree in general terms. However, the trend of the EOMCCSD and MRCCSD results is not the same. However, the trend of the two versions of the EOM CCSD results is the same, with the excited 1A 1 dipole moment values being the lowest. The EOM CCSD dipole moments calculated with the expectation value methods are lower than the finite field ones. Turning to the MRCC results, we observe that the values of the dipole moment are the smallest for the lowest A 2 state in both the singlet and triplet categories
Acknowledgements NV acknowledges the finantial support from Council of Scientific and Industrail Research ŽCSIR.. The authors acknowledge the Department of Science and Technology for funding. Dec Alpha 200 workstation was used for the computation of this work.
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References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x
J. Cizek, J. Chem. Phys. 45 Ž1966. 4156. R.J. Bartlett, Ann. Rev. Phys. Chem. 32 Ž1981. 359. H.J. Monkhorst, Int. J. Quantum Chem. S 11 Ž1977. 421. J. Noga, M. Urban, Theor. Chim. Acta 73 Ž1988. 291. H. Sekino, R.J. Bartlett, Int. J. Quantum Chem. S 18 Ž1984. 255. H. Koch, P. Jorgensen, J. Chem. Phys. 93 Ž1990. 3333. K.B. Ghose, P. Nair, S. Pal, Chem. Phys. Lett. 211 Ž1993. 15. N. Vaval, S. Pal, Phys. Rev. A 54 Ž1996. 250. S. Pal, Theor. Chim. Acta 66 Ž1984. 151. S. Pal, Phys. Rev. A 33 Ž1986. 2240. E.A. Salter, G.W. Trucks, R.J. Bartlett, J. Chem. Phys. 90 Ž1989. 1752. D. Mukherjee, S. Pal, Adv. Quantum Chem. 20 Ž1989. 291. W. Kutzelnigg, J. Chem. Phys. 77 Ž1982. 2081. I. Lindgren, Phys. Scr. 32 Ž1985. 291. D. Mukherjee, R.K. Moitra, A. Mukhopadhyay, Mol. Phys. 30 Ž1975. 1861. B. Jeziorski, H. Monkhorst, Phys. Rev. A 24 Ž1981. 1668.
w17x B. Jeziorski, J. Paldus, J. Chem. Phys. 8 Ž1988. 5673. w18x K.B. Ghose, S. Pal, J. Chem. Phys. 97 Ž1992. 3863. w19x J. Paldus, P. Piecuch, L. Pylypow, B. Jeziorski, Phys. Rev. A 47 Ž1993. 2738. w20x P. Durand, J.P. Malrieu, Adv. Chem. Phys. 67 Ž1987. 321. w21x S. Pal, M. Rittby, R.J. Bartlett, D. Sinha, D. Mukherjee, Chem. Phys. Lett. 137 Ž1987. 273. w22x N. Vaval, K.B. Ghose, S. Pal, D. Mukherjee, Chem. Phys. Lett. 209 Ž1993. 291. w23x S. Pal, M. Rittby, R.J. Bartlett, D. Sinha, D. Mukherjee, J. Chem. Phys. 88 Ž1988. 4357. w24x S. Pal, Int. J. Quantum Chem. 41 Ž1992. 443. w25x P. Szalay, Int. J. Quantum Chem. 55 Ž1995. 151. w26x D. Mukherjee, Chem. Phys. Lett. 125 Ž1986. 207. w27x A. Sadlej, Theor. Chim. Acta 79 Ž1991. 123. w28x J.F. Stanton, R.J. Bartlett, J. Chem. Phys. 98 Ž1993. 7029. w29x M. Urban, A.J. Sadlej, Theor. Chem. Acta 78 Ž1990. 189. w30x J.W.C. Johns, Can. J. Phys. 41 Ž1963. 209. w31x A. Chutjian, S. Trajmer, R.I. Hall, J. Chem. Phys. 63 Ž1975. 892. w32x F.W.E. Knoop, H.H. Brongersma, C.J. Oosterhoff, Chem. Phys. Lett. 13 Ž1972. 20. w33x G. Schulz, J. Chem. Phys. 33 Ž1960. 1661.