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Multireference Space Without First Solving the Configuration Interaction Problem Vitaly N. Glushkov,*[a] and Xavier Assfeld[b] We further develop an idea to generate a compact multireference space without first solving the configuration interaction problem previously proposed for the ground state (GS) (Glushkov, Chem. Phys. Lett. 1995, 244, 1). In the present contribution, our attention is focused on low-lying excited states (ESs) with the same symmetry as the GS which can be adequately described in terms of an high-spin open-shell formalism. Two references Mïller–Plesset (MP) like perturbation theory for ESs is developed. It is based on: (1) a main reference configuration constructed from the parent molecular orbitals adjusted to a given ES and (2) secondary double excitation configuration built on the GS like orbitals determined by the Hartree–Fock
equations subject to some orthogonality constraints. It is shown how to modify the MP zeroth-order Hamiltonian so that the reference configurations and corresponding excitations are eigenfunctions of it and are compatible with orthogonality conditions for the GS and ES. Intruder states appearance is also discussed. The proposed scheme is applied to the GS, ES, and excitation energies of small moleC 2013 Wiley cules to illustrate and calibrate our calculations. V Periodicals, Inc.
Introduction
Therefore, over past few decades, there has been a continuous effort to develop reliable and robust multireference (MR) methods for describing the correlation effects and the genuine MR-PTs have been proposed, for example, Refs. [7–9] (see also overviews[10–14] and bibliography therein). Nowadays, there are many MR-PT formulations which are focused on the different important issues of the theory exploiting both Rayleigh– €dinger[15–24] and Brillouin–Wigner schemes (see e.g., Schro Refs. [25–27] and also a website[28]) and their developments continuously are increasing. Especially, the so-called first-diagonalize-then-perturb MR methods have become rather popular.[15, 18, 19, 21] One of the essences of the development of MR-based theory is to use as small a reference space as possible.[13] In other words, a key moment is a choice of the MR space and simplicity to construct series of the MR many-body PT based on this subspace. Certainly, an opportunity to extend such approaches to the ES problem is very desirable. In addition, such an extension should lead to a balanced description of states and, therefore, reasonable excitation energies. The main goal of our work is to report one simple scheme (among all the possible ones) of generating the optimal and small MR spaces without first solving the configuration interaction (CI) problem whose solution is, in general, a time
It is well known that a single-reference Mïller–Plesset perturbation theory (MPPT) is one of the most popular computational tools in quantum chemistry due to combining the successful choice of a zeroth-order Hamiltonian to take the correlation effects into account and the effectiveness of the Hartree–Fock (HF) approximation. In cases where one configuration dominates, the single-determinant approximation can provide a sufficiently reliable basis for the development of many-particle methods that take into account correlation effects for both the ground state (GS) and excited state (ES),[1] especially if the corresponding reference function is optimized for a given state.[2–4] A similar situation takes place, in particular, in the description of ionized states caused by the excitation of valence electrons or core electrons, that is, states resulting from the removal of electrons from the valence spin orbital or the core spin orbital.[5, 6] Nevertheless, at the second-order MPPT level of calculations, the single, triple, quadruple, and so forth, configurational wave functions do not contribute to the correlation effects. A more complete account of electronic correlation is achieved at the level of higher order PT with more complicated and less feasible calculation schemes. A single-reference approximation restricts the accuracy of calculations, in particular, when the performance for bond-breaking problems is analyzed and perturbation treatment of ESs having the same symmetry as the GS is considered. A desire to extend the opportunity of MPPT to these situations while the advantages of the genuine method leads to the problem: How to generate supplementary configurations and modify the zeroth-order Hamiltonian so that these configurations would be its eigenfunctions and, therefore, the reduced resolvent operator has a diagonal form in such a many-particle basis set.
DOI: 10.1002/jcc.23502
[a] V. N. Glushkov Department of Physics, Electronics and Computer Systems, Oles Gonchar Dnipropetrovsk National University, Dnipropetrovsk, Ukraine E-mail: [email protected] [b] X. Assfeld Th eorie-Mod elisation-Simulation, UMR CNRS 7565, Facult e des Sciences et Technologies, Universit e de Lorraine, 54506, Vandoeuvre-le`s-Nancy, France C 2013 Wiley Periodicals, Inc. V
Journal of Computational Chemistry 2014, 35, 313–323
313
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consuming procedure. For the closed-shell systems, the basic idea of the method has already been given and tested for the GS s in a previous article.[29] In the present article, our attention is focused on low-lying ESs with the same symmetry as the GS which can be adequately described in terms of an high-spin open-shell formalism. We do not investigate the performance for bond-breaking problems here. At this stage of our study, one can say that, probably, we cannot expect that the method in question works properly for very large internuclear distances because the restricted HF-based methods are used to generate MR space. Nevertheless, we show below that the method provides reliable results for the GS and ES for distances up to R 5 2Re (Re is an equilibrium distance). To be selfcontained and to clarify some features of this work in the next section, a method of generating a new reference Slater determinant for the GS and some aspects that form the basis of the method are described. Second-order corrections based on the two-reference MP like PT are given for both closed-shell and high spin open-shell systems in the GSs. Instead of the popular first-diagonalize-then-perturb procedure, we first solve the amended HF problem then perturb. Section “Reference configurations and second-order energy expressions for excited states of open-shell systems” focuses the newer aspects of the method in question. In particular, orthogonality constrained HF equations for the parent orbitals optimized for a given ES are presented. It is shown how to modify the results of the previous section to develop the two-configuration PT for ESs having the same spatial and spin symmetry as the ground one. In addition, different forms for zeroth-order Hamiltonian based on the two-reference subspace and intruder state problem are discussed. Section “Test applications” contains discussions of test calculations and our conclusions.
Second Reference Configurations for GSs Usually, mutireference configuration selfconsistent-field (MCSCF) wave functions or complete active space SCF (CASSCF) method are used to form MR spaces. Therefore, a CI problem should be solved before selecting the most important configurations. In this section, following Ref. [29], a method of generating supplement reference configurations is briefly outlined. Neither MCSCF nor CASSCF wave functions are necessary in the application of this method to the correlation energy problem. The secondary double excitation configuration is constructed by optimizing the spin-orbitals placing orthogonality constraints to key spin-orbital from the first reference Slater determinant. Using a different spinorbital set from that of the main configuration for the secondary configuration, higher-order effects including the triple and more excitations can be incorporated. We shall show how to modify the MP zeroth-order Hamiltonian so that the singly, doubly, and so forth, excited configurations based on the new reference configuration are eigenfunctions of it. This makes it possible to develop two-reference PT which keeps the computational advantages of the genuine MP perturbation scheme.
314
Journal of Computational Chemistry 2014, 35, 313–323
Outline of a method for generating supplement reference configurations ð0Þ
Closed-shell systems. Let U0
be a single reference configuration of the GS which is assumed to be nondegenerate. It is constructed from doubly occupied spatial orbitals fu0i g; i51; 2; . . . ; n; which are determined by the HF equations: ðF0 2 e0i Þju0i i50; i51; 2; . . . ; n; n11; . . . ; M0
(1)
Here, F0 is the closed-shell Fock operator, M0 is the number of basis set functions, and 2n 5 N is the number of electrons. We shall now consider a method of generating a new reference configuration which is different from existing approaches based on a CI solution. Variational principles and experience accumulated by extensive calculation practices show the important role of double excitations. Therefore, we suggest minimizing a functional[29] E fWg5
hWjHjWi ; hWjWi
(2)
subject to orthogonality constraints hu0k j/i i50;
(3)
to build the second reference configuration W based on a new set of doubly occupied orbitals f/i g different from fu0i g. H is the system Hamiltonian having bound states. It is clear that the requirement (3) leads to orthogonality conditions for ð0Þ determinants hU0 jWi50: In addition, W looks like an optimal doubly excited configuration. In turn, singly, doubly, and so forth, excited configurations from W may be treated as higher order contributions to the correlation corrections. We shall show that these contributions can be included within a second-order two-reference MP like PT. The solution of (2) and (3) would be a complex problem if traditional methods of constraining optimization were used. In contrast, a simple to implement asymptotic projection (AP) method previously developed by us[30–33] can be applied to this problem with the same computational effort as for the GS eq. (1). This makes it possible to efficiently reduce the constrained problem to the unconstrained one. Starting from a stationary condition for the functional (2) and using the AP method a set of amended HF equations determining the new set of orbitals can be obtained F/ 1ko Pk 2ei j/i i50; ko ! 1; i51; 2; . . . ; n; n11; . . . ; M (4) with Pk 5ju0k ihu0k j. F/ is the standard closed-shell Fokian based on f/i g while F0 is based on {u0i }. According to the AP methodology,[31–33] the term ko Pk ; ko ! 1, in eq. (4) ensures the exact orthogonality. In general, different basis sets for conð0Þ figurations U0 and W may be used to improve calculation accuracy. Our experience showed that a choice of orbital u0k from ð0Þ U0 is not a trivial task. To fix the only second reference WWW.CHEMISTRYVIEWS.COM
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configuration we may, for example, use another variational principle ð0Þ
jhU0 jHjWij2 ð0Þ E0 2 E ð0Þ
5min
(5)
start from a stationary condition for the functional (2) under the orthogonality constraints hua0k j/ai i50; and hub0k j/bi i50:
(8) ð0Þ
which ensures the best energy lowering in the second order ð0Þ ð0Þ ð0Þ PT. In (5) E0 5 hU0 jHð0Þ jU0 i, E ð0Þ 5 hWjHð0Þ jWi, and Hð0Þ is the zeroth-order Hamiltonian (see below). The choice of u0k , which leads to a minimal energy hWjHjWi with respect to other occupied orbitals is crucial as it imposes a symmetry of the additional configurations. We shall show that the reference space constructed in accordance with eqs. (1) and (4) allows us to generalize the canonical variant of MPPT in a simple way. For the closed-shell systems, the expressions for second-order corrections can be obtained from the corresponding results for high-spin openshell systems which are outlined below. High-spin open shell systems. In the traditional Roothaan’s
open-shell method,[34] the occupied b spin orbitals are taken to be constructed from the same spatial orbitals as the a spin orbitals, that is, {ub0i } 5 {ua0i }. It is known that there is a degree of arbitrariness in the definition of the Fock operators based on this formalism that leads to different forms of the perturbation expansion for the correlation energy (see discussion in Refs. [35–39]). These ambiguities and the problem of off-diagonal Lagrangian multipliers are avoided in the AP open-shell methodology[40, 41] based on the unrestricted HF (UHF) formalism with some restrictions, that is, here we allow the spatial a orbitals to differ from those associated with b spin. However, these orbitals satisfy the correct variational conditions (generalized Brillouin’s theorem) and lead to spin-purity and the energy that is equivalent to that obtained by the Roothaan method. It should be noted that high-spin open-shell orbitals of our previous article[29] do not satisfy the generalized Brillouin’s theorem. ð0Þ The open-shell Slater determinant U0 is built from orbitals a a a u0i , i 5 1,2,. . ., n , associated with a spin and orbitals ub0i , ib 5 1,2,. . ., nb, associated with b spin, na > nb and n 5 na 1 nb is the total number of electrons. According to Refs. [40, 41], for the GS, we deal with the UHF like equations determining optimum set of orbitals ðF0a 2 ks Pb0 2 ea0i Þjua0i i50;
(6)
ks ! 1 ðF0b 1 ks Qa0 2 eb0i Þjub0i i50;
(7)
Here, Pb0 and Qa0 are orthoprojectors on the subspaces defined by occupied b-orbitals and virtual a-orbitals, respectively. In eqs. (6) and (7) F0a and F0b are the UHF operators. The additional terms. ks Pb0 and with ks Qa0 , ks ! 1 ensure that the spatial b spin orbitals are identical to those associated with a spinð0Þ purity for U0 . Generation of the second reference configuration can be considered as generalization of eqs. (2) and (3). Indeed, we
The ua0k and ub0k are the occupied orbitals in U0 . Then using the AP method we arrive at a set of HF like equations for the new orbitals f/i g: F/a 1ko Pak 2eai j/ai i50;
(9)
ko ! 1 F/b 1ko Pbk 2ebi j/bi i50
(10)
where Pak 5 jua0k ihua0k j and Pbk 5 jub0k ihub0k j. The Fockians F/a and F/b are based on f/i g. As a rule, the numbers ka and kb are the same. In this case, the magnitude hWjS2 jWi is closer to the true one and we deal with small spin contamination in the second reference function. In general, we can also impose on the f/i g spin-purity constraints. In this case similar to eqs. (6) and (7), the additional terms ks Pb and ks Qa should be added in eqs. (9) and (10), respectively. The corresponding determinant W agrees with an optimal doubly excited configuration and this result is transformed into the closed-shell one when na 5 nb . Thus, we can construct an optimal second reference function with ð0Þ essentially the same computational effort as U0 . In concluding this subsection we note: as neither kS nor kO can be infinity in practical calculations, one has to settle on some large finite values. On the one hand, the greater the value of kS or kO, the more accurately the respective constraint is satisfied. On the other hand, large values of kS and kO make it more difficult to converge for the SCF procedure. The recommended values used throughout this work are kS 5 100 hartrees for the spin-purity constraint and kO 5 1000 hartrees for the orthogonality constraint. Second-order PT corrections based on a single and two reference spaces. The above mentioned technique (subsection
“High-spin open shell systems”) provides a well-defined zeroorder approximation for open-shell single reference PT that ensures that single excitations do not contribute to the second-order energy.[42] This should be contrasted with the restricted Moller-Plesset (RMP)[35] method for which the generalized Brillouin’s theorem is not satisfied and consequently single replacement contributions enter the second-order energy expression. The zeroth-order Hamiltonian can be built as a sum of Fock operators for each electron: ð0Þ
H0 5
na X k
F0a ðk Þ1
nb X
F0b ðkÞ
(11)
k
with Fock operators Journal of Computational Chemistry 2014, 35, 313–323
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F0a 5
M0 X
F0b 5
jua0i iea0i hua0i j;
i
M0 X
jub0i ieb0i hub0i j;
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
H0 jU0 5 E0 jU0 i; E0 5 hU0 jHjU0 i5
na nb X X ea0i 1 eb0i i
ð0 Þ
i
ð0 Þ
H0 jUa0i i5 E0ia jUa0i i; E0ia 5 E0 2 ec0i 1 ec0a ; c5 a; b ð0 Þ
ð0 Þ
c c c c ab ab ab H0 jUab 0ij i5 E0ij jU0ij i; E0ij 5 E0 2 e0i 2e0j 1 e0a 1 e0b ;
(13) Note that the reduced resolvent operator ð0Þ
ð0 Þ
R0 5 Q0
ð0 Þ
ð0Þ 21
E0 2 H0
ð0 Þ
Q0 ;
ð0 Þ
ð0 Þ
ð0Þ
occ X virt X jUa ihUa j a
i
0i 0i ð0 Þ E0 2 E0ia
1
occ X virt jUab ihUab j X 0ij 0ij ð0Þ
i
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Multireference Space Without First Solving the Configuration Interaction Problem Vitaly N. Glushkov,*[a] and Xavier Assfeld[b] We further develop an idea to generate a compact multireference space without first solving the configuration interaction problem previously proposed for the ground state (GS) (Glushkov, Chem. Phys. Lett. 1995, 244, 1). In the present contribution, our attention is focused on low-lying excited states (ESs) with the same symmetry as the GS which can be adequately described in terms of an high-spin open-shell formalism. Two references Mïller–Plesset (MP) like perturbation theory for ESs is developed. It is based on: (1) a main reference configuration constructed from the parent molecular orbitals adjusted to a given ES and (2) secondary double excitation configuration built on the GS like orbitals determined by the Hartree–Fock
equations subject to some orthogonality constraints. It is shown how to modify the MP zeroth-order Hamiltonian so that the reference configurations and corresponding excitations are eigenfunctions of it and are compatible with orthogonality conditions for the GS and ES. Intruder states appearance is also discussed. The proposed scheme is applied to the GS, ES, and excitation energies of small moleC 2013 Wiley cules to illustrate and calibrate our calculations. V Periodicals, Inc.
Introduction
Therefore, over past few decades, there has been a continuous effort to develop reliable and robust multireference (MR) methods for describing the correlation effects and the genuine MR-PTs have been proposed, for example, Refs. [7–9] (see also overviews[10–14] and bibliography therein). Nowadays, there are many MR-PT formulations which are focused on the different important issues of the theory exploiting both Rayleigh– €dinger[15–24] and Brillouin–Wigner schemes (see e.g., Schro Refs. [25–27] and also a website[28]) and their developments continuously are increasing. Especially, the so-called first-diagonalize-then-perturb MR methods have become rather popular.[15, 18, 19, 21] One of the essences of the development of MR-based theory is to use as small a reference space as possible.[13] In other words, a key moment is a choice of the MR space and simplicity to construct series of the MR many-body PT based on this subspace. Certainly, an opportunity to extend such approaches to the ES problem is very desirable. In addition, such an extension should lead to a balanced description of states and, therefore, reasonable excitation energies. The main goal of our work is to report one simple scheme (among all the possible ones) of generating the optimal and small MR spaces without first solving the configuration interaction (CI) problem whose solution is, in general, a time
It is well known that a single-reference Mïller–Plesset perturbation theory (MPPT) is one of the most popular computational tools in quantum chemistry due to combining the successful choice of a zeroth-order Hamiltonian to take the correlation effects into account and the effectiveness of the Hartree–Fock (HF) approximation. In cases where one configuration dominates, the single-determinant approximation can provide a sufficiently reliable basis for the development of many-particle methods that take into account correlation effects for both the ground state (GS) and excited state (ES),[1] especially if the corresponding reference function is optimized for a given state.[2–4] A similar situation takes place, in particular, in the description of ionized states caused by the excitation of valence electrons or core electrons, that is, states resulting from the removal of electrons from the valence spin orbital or the core spin orbital.[5, 6] Nevertheless, at the second-order MPPT level of calculations, the single, triple, quadruple, and so forth, configurational wave functions do not contribute to the correlation effects. A more complete account of electronic correlation is achieved at the level of higher order PT with more complicated and less feasible calculation schemes. A single-reference approximation restricts the accuracy of calculations, in particular, when the performance for bond-breaking problems is analyzed and perturbation treatment of ESs having the same symmetry as the GS is considered. A desire to extend the opportunity of MPPT to these situations while the advantages of the genuine method leads to the problem: How to generate supplementary configurations and modify the zeroth-order Hamiltonian so that these configurations would be its eigenfunctions and, therefore, the reduced resolvent operator has a diagonal form in such a many-particle basis set.
DOI: 10.1002/jcc.23502
[a] V. N. Glushkov Department of Physics, Electronics and Computer Systems, Oles Gonchar Dnipropetrovsk National University, Dnipropetrovsk, Ukraine E-mail: [email protected] [b] X. Assfeld Th eorie-Mod elisation-Simulation, UMR CNRS 7565, Facult e des Sciences et Technologies, Universit e de Lorraine, 54506, Vandoeuvre-le`s-Nancy, France C 2013 Wiley Periodicals, Inc. V
Journal of Computational Chemistry 2014, 35, 313–323
313
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consuming procedure. For the closed-shell systems, the basic idea of the method has already been given and tested for the GS s in a previous article.[29] In the present article, our attention is focused on low-lying ESs with the same symmetry as the GS which can be adequately described in terms of an high-spin open-shell formalism. We do not investigate the performance for bond-breaking problems here. At this stage of our study, one can say that, probably, we cannot expect that the method in question works properly for very large internuclear distances because the restricted HF-based methods are used to generate MR space. Nevertheless, we show below that the method provides reliable results for the GS and ES for distances up to R 5 2Re (Re is an equilibrium distance). To be selfcontained and to clarify some features of this work in the next section, a method of generating a new reference Slater determinant for the GS and some aspects that form the basis of the method are described. Second-order corrections based on the two-reference MP like PT are given for both closed-shell and high spin open-shell systems in the GSs. Instead of the popular first-diagonalize-then-perturb procedure, we first solve the amended HF problem then perturb. Section “Reference configurations and second-order energy expressions for excited states of open-shell systems” focuses the newer aspects of the method in question. In particular, orthogonality constrained HF equations for the parent orbitals optimized for a given ES are presented. It is shown how to modify the results of the previous section to develop the two-configuration PT for ESs having the same spatial and spin symmetry as the ground one. In addition, different forms for zeroth-order Hamiltonian based on the two-reference subspace and intruder state problem are discussed. Section “Test applications” contains discussions of test calculations and our conclusions.
Second Reference Configurations for GSs Usually, mutireference configuration selfconsistent-field (MCSCF) wave functions or complete active space SCF (CASSCF) method are used to form MR spaces. Therefore, a CI problem should be solved before selecting the most important configurations. In this section, following Ref. [29], a method of generating supplement reference configurations is briefly outlined. Neither MCSCF nor CASSCF wave functions are necessary in the application of this method to the correlation energy problem. The secondary double excitation configuration is constructed by optimizing the spin-orbitals placing orthogonality constraints to key spin-orbital from the first reference Slater determinant. Using a different spinorbital set from that of the main configuration for the secondary configuration, higher-order effects including the triple and more excitations can be incorporated. We shall show how to modify the MP zeroth-order Hamiltonian so that the singly, doubly, and so forth, excited configurations based on the new reference configuration are eigenfunctions of it. This makes it possible to develop two-reference PT which keeps the computational advantages of the genuine MP perturbation scheme.
314
Journal of Computational Chemistry 2014, 35, 313–323
Outline of a method for generating supplement reference configurations ð0Þ
Closed-shell systems. Let U0
be a single reference configuration of the GS which is assumed to be nondegenerate. It is constructed from doubly occupied spatial orbitals fu0i g; i51; 2; . . . ; n; which are determined by the HF equations: ðF0 2 e0i Þju0i i50; i51; 2; . . . ; n; n11; . . . ; M0
(1)
Here, F0 is the closed-shell Fock operator, M0 is the number of basis set functions, and 2n 5 N is the number of electrons. We shall now consider a method of generating a new reference configuration which is different from existing approaches based on a CI solution. Variational principles and experience accumulated by extensive calculation practices show the important role of double excitations. Therefore, we suggest minimizing a functional[29] E fWg5
hWjHjWi ; hWjWi
(2)
subject to orthogonality constraints hu0k j/i i50;
(3)
to build the second reference configuration W based on a new set of doubly occupied orbitals f/i g different from fu0i g. H is the system Hamiltonian having bound states. It is clear that the requirement (3) leads to orthogonality conditions for ð0Þ determinants hU0 jWi50: In addition, W looks like an optimal doubly excited configuration. In turn, singly, doubly, and so forth, excited configurations from W may be treated as higher order contributions to the correlation corrections. We shall show that these contributions can be included within a second-order two-reference MP like PT. The solution of (2) and (3) would be a complex problem if traditional methods of constraining optimization were used. In contrast, a simple to implement asymptotic projection (AP) method previously developed by us[30–33] can be applied to this problem with the same computational effort as for the GS eq. (1). This makes it possible to efficiently reduce the constrained problem to the unconstrained one. Starting from a stationary condition for the functional (2) and using the AP method a set of amended HF equations determining the new set of orbitals can be obtained F/ 1ko Pk 2ei j/i i50; ko ! 1; i51; 2; . . . ; n; n11; . . . ; M (4) with Pk 5ju0k ihu0k j. F/ is the standard closed-shell Fokian based on f/i g while F0 is based on {u0i }. According to the AP methodology,[31–33] the term ko Pk ; ko ! 1, in eq. (4) ensures the exact orthogonality. In general, different basis sets for conð0Þ figurations U0 and W may be used to improve calculation accuracy. Our experience showed that a choice of orbital u0k from ð0Þ U0 is not a trivial task. To fix the only second reference WWW.CHEMISTRYVIEWS.COM
FULL PAPER
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configuration we may, for example, use another variational principle ð0Þ
jhU0 jHjWij2 ð0Þ E0 2 E ð0Þ
5min
(5)
start from a stationary condition for the functional (2) under the orthogonality constraints hua0k j/ai i50; and hub0k j/bi i50:
(8) ð0Þ
which ensures the best energy lowering in the second order ð0Þ ð0Þ ð0Þ PT. In (5) E0 5 hU0 jHð0Þ jU0 i, E ð0Þ 5 hWjHð0Þ jWi, and Hð0Þ is the zeroth-order Hamiltonian (see below). The choice of u0k , which leads to a minimal energy hWjHjWi with respect to other occupied orbitals is crucial as it imposes a symmetry of the additional configurations. We shall show that the reference space constructed in accordance with eqs. (1) and (4) allows us to generalize the canonical variant of MPPT in a simple way. For the closed-shell systems, the expressions for second-order corrections can be obtained from the corresponding results for high-spin openshell systems which are outlined below. High-spin open shell systems. In the traditional Roothaan’s
open-shell method,[34] the occupied b spin orbitals are taken to be constructed from the same spatial orbitals as the a spin orbitals, that is, {ub0i } 5 {ua0i }. It is known that there is a degree of arbitrariness in the definition of the Fock operators based on this formalism that leads to different forms of the perturbation expansion for the correlation energy (see discussion in Refs. [35–39]). These ambiguities and the problem of off-diagonal Lagrangian multipliers are avoided in the AP open-shell methodology[40, 41] based on the unrestricted HF (UHF) formalism with some restrictions, that is, here we allow the spatial a orbitals to differ from those associated with b spin. However, these orbitals satisfy the correct variational conditions (generalized Brillouin’s theorem) and lead to spin-purity and the energy that is equivalent to that obtained by the Roothaan method. It should be noted that high-spin open-shell orbitals of our previous article[29] do not satisfy the generalized Brillouin’s theorem. ð0Þ The open-shell Slater determinant U0 is built from orbitals a a a u0i , i 5 1,2,. . ., n , associated with a spin and orbitals ub0i , ib 5 1,2,. . ., nb, associated with b spin, na > nb and n 5 na 1 nb is the total number of electrons. According to Refs. [40, 41], for the GS, we deal with the UHF like equations determining optimum set of orbitals ðF0a 2 ks Pb0 2 ea0i Þjua0i i50;
(6)
ks ! 1 ðF0b 1 ks Qa0 2 eb0i Þjub0i i50;
(7)
Here, Pb0 and Qa0 are orthoprojectors on the subspaces defined by occupied b-orbitals and virtual a-orbitals, respectively. In eqs. (6) and (7) F0a and F0b are the UHF operators. The additional terms. ks Pb0 and with ks Qa0 , ks ! 1 ensure that the spatial b spin orbitals are identical to those associated with a spinð0Þ purity for U0 . Generation of the second reference configuration can be considered as generalization of eqs. (2) and (3). Indeed, we
The ua0k and ub0k are the occupied orbitals in U0 . Then using the AP method we arrive at a set of HF like equations for the new orbitals f/i g: F/a 1ko Pak 2eai j/ai i50;
(9)
ko ! 1 F/b 1ko Pbk 2ebi j/bi i50
(10)
where Pak 5 jua0k ihua0k j and Pbk 5 jub0k ihub0k j. The Fockians F/a and F/b are based on f/i g. As a rule, the numbers ka and kb are the same. In this case, the magnitude hWjS2 jWi is closer to the true one and we deal with small spin contamination in the second reference function. In general, we can also impose on the f/i g spin-purity constraints. In this case similar to eqs. (6) and (7), the additional terms ks Pb and ks Qa should be added in eqs. (9) and (10), respectively. The corresponding determinant W agrees with an optimal doubly excited configuration and this result is transformed into the closed-shell one when na 5 nb . Thus, we can construct an optimal second reference function with ð0Þ essentially the same computational effort as U0 . In concluding this subsection we note: as neither kS nor kO can be infinity in practical calculations, one has to settle on some large finite values. On the one hand, the greater the value of kS or kO, the more accurately the respective constraint is satisfied. On the other hand, large values of kS and kO make it more difficult to converge for the SCF procedure. The recommended values used throughout this work are kS 5 100 hartrees for the spin-purity constraint and kO 5 1000 hartrees for the orthogonality constraint. Second-order PT corrections based on a single and two reference spaces. The above mentioned technique (subsection
“High-spin open shell systems”) provides a well-defined zeroorder approximation for open-shell single reference PT that ensures that single excitations do not contribute to the second-order energy.[42] This should be contrasted with the restricted Moller-Plesset (RMP)[35] method for which the generalized Brillouin’s theorem is not satisfied and consequently single replacement contributions enter the second-order energy expression. The zeroth-order Hamiltonian can be built as a sum of Fock operators for each electron: ð0Þ
H0 5
na X k
F0a ðk Þ1
nb X
F0b ðkÞ
(11)
k
with Fock operators Journal of Computational Chemistry 2014, 35, 313–323
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F0a 5
M0 X
F0b 5
jua0i iea0i hua0i j;
i
M0 X
jub0i ieb0i hub0i j;
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
H0 jU0 5 E0 jU0 i; E0 5 hU0 jHjU0 i5
na nb X X ea0i 1 eb0i i
ð0 Þ
i
ð0 Þ
H0 jUa0i i5 E0ia jUa0i i; E0ia 5 E0 2 ec0i 1 ec0a ; c5 a; b ð0 Þ
ð0 Þ
c c c c ab ab ab H0 jUab 0ij i5 E0ij jU0ij i; E0ij 5 E0 2 e0i 2e0j 1 e0a 1 e0b ;
(13) Note that the reduced resolvent operator ð0Þ
ð0 Þ
R0 5 Q0
ð0 Þ
ð0Þ 21
E0 2 H0
ð0 Þ
Q0 ;
ð0 Þ
ð0 Þ
ð0Þ
occ X virt X jUa ihUa j a
i
0i 0i ð0 Þ E0 2 E0ia
1
occ X virt jUab ihUab j X 0ij 0ij ð0Þ
i
