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The use of polarized atomic orbitals in the initial guess in mcscf calculations
1
Journal of Molecular Structure (Theo&em), 208 (1990) 1-6 Elsevier Science Publishers B.V., Amsterdam
THE USE OF POLARIZED ATOMIC GUESS IN MCSCF CALCULATIONS
ORBITALS
IN THE INITIAL
ALFRED0 M.J. SANCHEZ DE MERAS, FRANCISCO TORRENS ZARAGOZA AND IGNACIO NEBOT GIL Departament de Quimica Fisica, Universitat de ValBncia, Dr. Moliner SO,46100 Burjassot, Valhzcia (Spain) (Received 4 August 1989)
ABSTRACT Sample calculations were made on small molecular systems in order to compare the efficiency of polarized atomic orbitals (PAOs) with Hartree-Fock ones as the initial guess in MCSCF calculations. In conclusion, we propose the use of the PA0 technique to improve the MCSCF convergence.
INTRODUCTION
It is well known that canonical Hartree-Fock (HF) orbitals are not well suited to the construction of the Slater determinants to be used in a CI expansion [ 11. In addition, the canonical occupied HF orbit& are almost the same as those with higher occupation numbers in an MCSCF calculation, and so the main effect of the multiconfigurational. expansion is that a unitary transformation is made among the virtual (in the HF sense) orbitals [ 21. Therefore, great effort has been directed to finding auxiliary algorithms which improve virtual orbitals in a cheaper way than do MCSCF calculations. An almost exhaustive review of such methods is given in Ref. 1. The same reasons that justify the inadequacy of HF orbitals in configuration interaction calculations apply to this type of orbitals and explain why it is not useful in making the initial guess in MCSCF calculations. It seems, therefore, convenient to improve them, especially inside the valence space. In this sense, the polarized atomic orbital (PAO) method of L&y and coworkers [ 3 ] seems to be reliable. PAOs are built up by projecting the HF atomic orbitals of the free atoms over the SCF virtual space of the molecule. In this way, the valence virtual molecular orbitals (MOs) are expanded as linear combinations of the valence atomic orbitals and, consequently, they become spatially concentrated. As a matter of fact, this is rather a simple projection procedure and for
0166-1266/90/$03.50
0 1990 -
Elsevier Science Publishers B.V.
2
that reason the acronym has also been interpreted [l] as projected atomic orbitals. We propose here the use of the PA0 technique to improve the MCSCF convergence and we compare the efficiency of these orbitals with the HF ones. Recently, the use of second-order Msller-Plesset natural orbitals (MP2 NOs) has been suggested [ 41 as a valid alternative to the traditional HF initial guess in MCSCF calculations, the advantage of PAOs resting on their easier determination. CALCULATION DETAILS
All calculations were done at the experimental equilibrium geometry of the molecular system considered [ 51 and in the Czv symmetry group even in the case of the C,, hydroxyl radical. The 6-31G** basis set [6] was also used in all calculations, employing in all cases the Cartesian six-component d orbitals. The additional s function so included was never deleted. Both SCF and MCSCF calculations were carried out using the SIRIUS program [ 71. This program uses the electronic integrals calculated by Almlijf’s MOLECULE [ 81 and the CI coupling elements generated by the GUGA code of Siegbahn [91. The PAOs were constructed by means of a program from the Laboratoire de Physique Quantique de la Universite Paul Sabatier de Toulouse [lo]. Finally, a small algorithm was developed to interface SIRIUS and the PA0 program. For the molecules studied, execution of the adapting interface program took about 20 s on a VAX-8300 computer. RESULTS AND DISCUSSION
The results of the convergence characteristics of the calculations are summarized in Tables l-4. Since calculations were carried out using two computers TABLE 1 MCSCF sample calculations for wateld Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.100071835 - 76.14126478 0.090004801
0.127788919 - 76.14126478 0.000006886
No. macroiterations No. microiterations CPU time(s)
16 106 2569.92
12 62 2009.94
“Complete active space built up by distributing eight electrons in (4~2 Q?n,); this implies 492 configurations and 53 orbital rotations.
3 TABLE 2 MCSCF sample calculations for CO (2)” Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.16477603 - 112.87646235 0.00009968
0.20174798 - 130.0051756Sb 0.48677147
No. macroiterations No. microiterations CPU time(s)
17 109 3793.19
14 116 2994.02
“Complete active space built up by distributing eight electrons in (4a2n$Zrc,); this implies 492 configurations and 91 orbital rotations. bCalculation not converged.
TABLE 3 MCSCF sample calculations for CO (3 )” Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.16807759 - 112.88211039 0.00009988
0.20483560 - 112.88211039 0.00006018
No. macroiterations No. microiterations CPU time(s)
14 71 7795.24
11 64 6836.41
“Complete active space built up by distributing ten electrons in (5a2nz2z,,); this implies 1436 configurations and 89 orbital rotations.
TABLE 4 MCSCF sample calculations for OH’ Initial guess
Hartree-Fock
PA0
PA0 + TC
Initial orbital gradient Final energy Final gradient
0.21276012 - 75.37986205 0.00008191
0.21987320 - 75.40421775 0.00006694
0.21987320 -75.40421775 0.00008021
No. macroiterations No. microiterations CPU time (9)
35 357 1822.29
19 86 933.92
8 42 342.33
“Complete active space built up by distributing seven electrons in (3u17rJ7rY);this implies 10 configurations and 37 orbital rotations.
4
of different velocity (VAX 8300 (0.15 Mflops) and VAXstation 3200 (0.41 Mflops) [ 111, the CPU times cited in the tables have been corrected to the first of the computers where necessary. In Table 4 PA0 +TC means that a more restrictive criterion has been chosen when checking the quadraticity of the MCSCF parameters hypersurface. In particular, the criterion for the optimization of excited states given in Ref. 12 was used. In this way divergence to pseudo-minima with a too large stabilization energy is avoided. This behaviour has also been found [ 131 when using the intermediate orbital optimization of Ref. 14 in open-shell systems. The first point to note in all cases is that the orbital gradient norm (see tables) is greater when PAOs are used in the initial guess for the MCSCF calculation. To account for this fact, let us remember that the gradient norm only decreases monotonically in the proximity of the desired stationary point, when total convergence is nearly reached. Therefore, in the calculations where HF orbitals were employed the gradient norm becomes about 8 a.u. in the second or third macroiteration. (We use the same nomenclature as that in Ref. 14 and references therein.) This large increment of gradient norm is due mainly to the orbital part of the MCSCF expansion and can be understood as a result of the complete inadequacy of the molecular orbitals to the actual CI coefficients in the first steps of the optimization. PAOs are better conditioned and, therefore, the fluctuations produced by optimization of the CI coefficients in respect to poor MOs are avoided. The results presented in Tables l-4 show that the improvement due to the use of PAOs is appreciable, with an important reduction in the number of both macroiterations and microiterations. Let us remember that each macroiteration requires an integral transformation to the molecular basis (and perhaps the determination of the natural orbitala for the present CI expansion [ 141) and, therefore, a lower number of macroiterations involves a considerable saving in CPU time. However, as fewer integral transformations are needed, input/output between the available core and external storage is also diminished, which gives rise to better use of computer resources and thereby decreases the time required for calculation. Decreasing the number of microiterations leads to fewer linear transformations with very large matrices. As a consequence, page faults in the virtual memory are sensibly reduced, and there is a further saving in the time required for the process. These improvements became larger the more similar is the chosen complete active space (CAS) to the one built by joining the valence space of the separated atoms, as can be seen from the comparison of the results for CO (Tables 2 and 3). In the largest space (the most similar to the best) the advantages of using PAOs are better shown. However, the use of PAOs seems not to be convenient in the smaller CAS in Table 2. Nevertheless, in this case, as well as in other similar ones not presented here, a faster convergence can be achieved by
5
using the more restrictive criterion mentioned previously. Unfortunately, such a criterion implies that some steps with a small, although acceptable, trust ratio may be rejected, thereby increasing unnecessarily the number of macroiterations. For the hydroxyl radical the use of MP2 NOs requires the solution of the rather complex open-shell RHF algorithm in such a manner that the saving in CPU time is offset by the difficulty in solving the above described problem. If PAOs are used as the initial guess, there is no major disadvantage when working with open-shell systems. The results of the calculation on this system clearly show the convenience of employing PAOs, even when the chosen active space is far from the optimal one. As can be seen from Table 4, the use of PAOs allows the calculation to be done in just over half the time necessary if HF orbitals are used. Furthermore, when tight control of the quadraticity of the hypersurface is imposed, the computation time is reduced to about 20% of the HF time. Another problem that arises in the MCSCF calculation with HF orbitals is that symmetric orthogonalization diverges after 20 macroiterations, and the user is then forced to employ the always more complicated Gram-Schmidt procedure. Finally the most serious problem is that important terms of the correlation energy were lost when using HF orbitals. This led to a mathematical artifact which causes the ground state to be confused with an excited one. This phenomenon has been detected previously by Jensen et al. [ 41 in their theoretical study of HCl. CONCLUSIONS
In summary, it can be said that the use of PAOs as the initial guess in MCSCF calculations is a valid choice and is better than using HF molecular orbitals. In the case of PAOs the use of a tight control of the step results is convenient, even taking into account the implicit disadvantages that their use may suppose; anyway, this problem can be overcome with a better definition of the control criterion. As a matter of fact, optimization of the virtual orbitals according to the exchange integral may be more adequate [ 151. Nevertheless, it should not be forgotten that exchange optimization is too difficult to be useful. A rigorous optimization of the exchange integral requires its maximization between the occupied canonical orbitals and the orbital correlating each one of them, as well as a minimization of the exchange among the valence occupied orbitals themselves. Mathematical methods designed to accomplish such a goal are rather scarce and difficult to work with, especially when dealing with largedimension matrices. Therefore, the use of PAOs is proposed as a good choice as the initial orbital guess in MCSCF calculations.
6 ACKNOWLEDGEMENTS
The authors are indebted to the members of the theoretical group at the Kemisk Institut of Arhus Universitet for providing the programs used for MCSCF calculations. Two of us, A.S.M. and F.T.Z., thank the Conselleria de Cultura Educaci6 i Cihncia de la Generalitat Valenciana for a grant.
REFERENCES 1 2 3 4 5 6
7 8 9 10 11 12 13 14 15
See, for example, F. Ihas, M. Merch&n, M. Pelissier and J.P. Malrieu, Chem. Phys., 107 (1986) 361 and references therein. B. L&y, Int. J. Quantum Chem., 4 (1970) 297. G. Chambaud, M. Gerard-Ain, E. Kassab, B. L&y and P. Pernot, Chem. Phys., 90 (1984) 271. H.J.Aa. Jensen, P. Jorgensen, H. &en and J. Olsen, J. Chem. Phys., 88 (1988) 3834. G. Hertzberg, Electronic Spectra of Polyatomic Molecules, Van Nostrand, Princeton, NJ, 1966. M. Frich, J.S. Binkley, H.B. Schegel, K. Raghavachari, R. Martin, J.J.P. Stewart, F. Bobrowicz, D. Defrees, R. Seeger, R. Whiteside, D. Fox, E. Fluder and J.A. Pople, GAUSSIAN 86, Release C. Carnegie Mellon University, Pittsburgh, PA, 1986. H.J.Aa. Jensen and H. &ren, Chem. Phys., 104 (1986) 229. J. Almlof, USIP 74-29, Stockholm University, 1974. P.E.M. Siegbahn, Chem. Phys. Lett., 109 (1984) 417. M. Pelissier, private communication. J.J. Dongarra, Technical Memorandum No. 23, Argonne National Laboratory, Argonne, IL, 1988. H.J.Aa. Jensen and P. Jergensen, J. Chem. Phys., 80 (1984) 1204. A. Sanchez de Me&, Ph.D. Thesis, Ulniversidad de Valencia, Valencia, 1989. H.J.Aa. Jensen, P. Jergensen and H. Agren, J. Chem. Phys., 87 (1987) 451. C.F. Bender andE.R. Davidson, J. Chem. Phys., 47 (1967) 4972.D. Feller andE.R. Davidson, J. Chem. Phys., 74 (1981) 3977. P. Fantucci, V. Bonacic-Koutecky and J. Koutecky, J. Comput. Chem., 6 (1985) 462.
Journal of Molecular Structure (Theo&em), 208 (1990) 1-6 Elsevier Science Publishers B.V., Amsterdam
THE USE OF POLARIZED ATOMIC GUESS IN MCSCF CALCULATIONS
ORBITALS
IN THE INITIAL
ALFRED0 M.J. SANCHEZ DE MERAS, FRANCISCO TORRENS ZARAGOZA AND IGNACIO NEBOT GIL Departament de Quimica Fisica, Universitat de ValBncia, Dr. Moliner SO,46100 Burjassot, Valhzcia (Spain) (Received 4 August 1989)
ABSTRACT Sample calculations were made on small molecular systems in order to compare the efficiency of polarized atomic orbitals (PAOs) with Hartree-Fock ones as the initial guess in MCSCF calculations. In conclusion, we propose the use of the PA0 technique to improve the MCSCF convergence.
INTRODUCTION
It is well known that canonical Hartree-Fock (HF) orbitals are not well suited to the construction of the Slater determinants to be used in a CI expansion [ 11. In addition, the canonical occupied HF orbit& are almost the same as those with higher occupation numbers in an MCSCF calculation, and so the main effect of the multiconfigurational. expansion is that a unitary transformation is made among the virtual (in the HF sense) orbitals [ 21. Therefore, great effort has been directed to finding auxiliary algorithms which improve virtual orbitals in a cheaper way than do MCSCF calculations. An almost exhaustive review of such methods is given in Ref. 1. The same reasons that justify the inadequacy of HF orbitals in configuration interaction calculations apply to this type of orbitals and explain why it is not useful in making the initial guess in MCSCF calculations. It seems, therefore, convenient to improve them, especially inside the valence space. In this sense, the polarized atomic orbital (PAO) method of L&y and coworkers [ 3 ] seems to be reliable. PAOs are built up by projecting the HF atomic orbitals of the free atoms over the SCF virtual space of the molecule. In this way, the valence virtual molecular orbitals (MOs) are expanded as linear combinations of the valence atomic orbitals and, consequently, they become spatially concentrated. As a matter of fact, this is rather a simple projection procedure and for
0166-1266/90/$03.50
0 1990 -
Elsevier Science Publishers B.V.
2
that reason the acronym has also been interpreted [l] as projected atomic orbitals. We propose here the use of the PA0 technique to improve the MCSCF convergence and we compare the efficiency of these orbitals with the HF ones. Recently, the use of second-order Msller-Plesset natural orbitals (MP2 NOs) has been suggested [ 41 as a valid alternative to the traditional HF initial guess in MCSCF calculations, the advantage of PAOs resting on their easier determination. CALCULATION DETAILS
All calculations were done at the experimental equilibrium geometry of the molecular system considered [ 51 and in the Czv symmetry group even in the case of the C,, hydroxyl radical. The 6-31G** basis set [6] was also used in all calculations, employing in all cases the Cartesian six-component d orbitals. The additional s function so included was never deleted. Both SCF and MCSCF calculations were carried out using the SIRIUS program [ 71. This program uses the electronic integrals calculated by Almlijf’s MOLECULE [ 81 and the CI coupling elements generated by the GUGA code of Siegbahn [91. The PAOs were constructed by means of a program from the Laboratoire de Physique Quantique de la Universite Paul Sabatier de Toulouse [lo]. Finally, a small algorithm was developed to interface SIRIUS and the PA0 program. For the molecules studied, execution of the adapting interface program took about 20 s on a VAX-8300 computer. RESULTS AND DISCUSSION
The results of the convergence characteristics of the calculations are summarized in Tables l-4. Since calculations were carried out using two computers TABLE 1 MCSCF sample calculations for wateld Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.100071835 - 76.14126478 0.090004801
0.127788919 - 76.14126478 0.000006886
No. macroiterations No. microiterations CPU time(s)
16 106 2569.92
12 62 2009.94
“Complete active space built up by distributing eight electrons in (4~2 Q?n,); this implies 492 configurations and 53 orbital rotations.
3 TABLE 2 MCSCF sample calculations for CO (2)” Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.16477603 - 112.87646235 0.00009968
0.20174798 - 130.0051756Sb 0.48677147
No. macroiterations No. microiterations CPU time(s)
17 109 3793.19
14 116 2994.02
“Complete active space built up by distributing eight electrons in (4a2n$Zrc,); this implies 492 configurations and 91 orbital rotations. bCalculation not converged.
TABLE 3 MCSCF sample calculations for CO (3 )” Initial guess
Hartree-Fock
PA0
Initial orbital gradient Final energy Final gradient norm
0.16807759 - 112.88211039 0.00009988
0.20483560 - 112.88211039 0.00006018
No. macroiterations No. microiterations CPU time(s)
14 71 7795.24
11 64 6836.41
“Complete active space built up by distributing ten electrons in (5a2nz2z,,); this implies 1436 configurations and 89 orbital rotations.
TABLE 4 MCSCF sample calculations for OH’ Initial guess
Hartree-Fock
PA0
PA0 + TC
Initial orbital gradient Final energy Final gradient
0.21276012 - 75.37986205 0.00008191
0.21987320 - 75.40421775 0.00006694
0.21987320 -75.40421775 0.00008021
No. macroiterations No. microiterations CPU time (9)
35 357 1822.29
19 86 933.92
8 42 342.33
“Complete active space built up by distributing seven electrons in (3u17rJ7rY);this implies 10 configurations and 37 orbital rotations.
4
of different velocity (VAX 8300 (0.15 Mflops) and VAXstation 3200 (0.41 Mflops) [ 111, the CPU times cited in the tables have been corrected to the first of the computers where necessary. In Table 4 PA0 +TC means that a more restrictive criterion has been chosen when checking the quadraticity of the MCSCF parameters hypersurface. In particular, the criterion for the optimization of excited states given in Ref. 12 was used. In this way divergence to pseudo-minima with a too large stabilization energy is avoided. This behaviour has also been found [ 131 when using the intermediate orbital optimization of Ref. 14 in open-shell systems. The first point to note in all cases is that the orbital gradient norm (see tables) is greater when PAOs are used in the initial guess for the MCSCF calculation. To account for this fact, let us remember that the gradient norm only decreases monotonically in the proximity of the desired stationary point, when total convergence is nearly reached. Therefore, in the calculations where HF orbitals were employed the gradient norm becomes about 8 a.u. in the second or third macroiteration. (We use the same nomenclature as that in Ref. 14 and references therein.) This large increment of gradient norm is due mainly to the orbital part of the MCSCF expansion and can be understood as a result of the complete inadequacy of the molecular orbitals to the actual CI coefficients in the first steps of the optimization. PAOs are better conditioned and, therefore, the fluctuations produced by optimization of the CI coefficients in respect to poor MOs are avoided. The results presented in Tables l-4 show that the improvement due to the use of PAOs is appreciable, with an important reduction in the number of both macroiterations and microiterations. Let us remember that each macroiteration requires an integral transformation to the molecular basis (and perhaps the determination of the natural orbitala for the present CI expansion [ 141) and, therefore, a lower number of macroiterations involves a considerable saving in CPU time. However, as fewer integral transformations are needed, input/output between the available core and external storage is also diminished, which gives rise to better use of computer resources and thereby decreases the time required for calculation. Decreasing the number of microiterations leads to fewer linear transformations with very large matrices. As a consequence, page faults in the virtual memory are sensibly reduced, and there is a further saving in the time required for the process. These improvements became larger the more similar is the chosen complete active space (CAS) to the one built by joining the valence space of the separated atoms, as can be seen from the comparison of the results for CO (Tables 2 and 3). In the largest space (the most similar to the best) the advantages of using PAOs are better shown. However, the use of PAOs seems not to be convenient in the smaller CAS in Table 2. Nevertheless, in this case, as well as in other similar ones not presented here, a faster convergence can be achieved by
5
using the more restrictive criterion mentioned previously. Unfortunately, such a criterion implies that some steps with a small, although acceptable, trust ratio may be rejected, thereby increasing unnecessarily the number of macroiterations. For the hydroxyl radical the use of MP2 NOs requires the solution of the rather complex open-shell RHF algorithm in such a manner that the saving in CPU time is offset by the difficulty in solving the above described problem. If PAOs are used as the initial guess, there is no major disadvantage when working with open-shell systems. The results of the calculation on this system clearly show the convenience of employing PAOs, even when the chosen active space is far from the optimal one. As can be seen from Table 4, the use of PAOs allows the calculation to be done in just over half the time necessary if HF orbitals are used. Furthermore, when tight control of the quadraticity of the hypersurface is imposed, the computation time is reduced to about 20% of the HF time. Another problem that arises in the MCSCF calculation with HF orbitals is that symmetric orthogonalization diverges after 20 macroiterations, and the user is then forced to employ the always more complicated Gram-Schmidt procedure. Finally the most serious problem is that important terms of the correlation energy were lost when using HF orbitals. This led to a mathematical artifact which causes the ground state to be confused with an excited one. This phenomenon has been detected previously by Jensen et al. [ 41 in their theoretical study of HCl. CONCLUSIONS
In summary, it can be said that the use of PAOs as the initial guess in MCSCF calculations is a valid choice and is better than using HF molecular orbitals. In the case of PAOs the use of a tight control of the step results is convenient, even taking into account the implicit disadvantages that their use may suppose; anyway, this problem can be overcome with a better definition of the control criterion. As a matter of fact, optimization of the virtual orbitals according to the exchange integral may be more adequate [ 151. Nevertheless, it should not be forgotten that exchange optimization is too difficult to be useful. A rigorous optimization of the exchange integral requires its maximization between the occupied canonical orbitals and the orbital correlating each one of them, as well as a minimization of the exchange among the valence occupied orbitals themselves. Mathematical methods designed to accomplish such a goal are rather scarce and difficult to work with, especially when dealing with largedimension matrices. Therefore, the use of PAOs is proposed as a good choice as the initial orbital guess in MCSCF calculations.
6 ACKNOWLEDGEMENTS
The authors are indebted to the members of the theoretical group at the Kemisk Institut of Arhus Universitet for providing the programs used for MCSCF calculations. Two of us, A.S.M. and F.T.Z., thank the Conselleria de Cultura Educaci6 i Cihncia de la Generalitat Valenciana for a grant.
REFERENCES 1 2 3 4 5 6
7 8 9 10 11 12 13 14 15
See, for example, F. Ihas, M. Merch&n, M. Pelissier and J.P. Malrieu, Chem. Phys., 107 (1986) 361 and references therein. B. L&y, Int. J. Quantum Chem., 4 (1970) 297. G. Chambaud, M. Gerard-Ain, E. Kassab, B. L&y and P. Pernot, Chem. Phys., 90 (1984) 271. H.J.Aa. Jensen, P. Jorgensen, H. &en and J. Olsen, J. Chem. Phys., 88 (1988) 3834. G. Hertzberg, Electronic Spectra of Polyatomic Molecules, Van Nostrand, Princeton, NJ, 1966. M. Frich, J.S. Binkley, H.B. Schegel, K. Raghavachari, R. Martin, J.J.P. Stewart, F. Bobrowicz, D. Defrees, R. Seeger, R. Whiteside, D. Fox, E. Fluder and J.A. Pople, GAUSSIAN 86, Release C. Carnegie Mellon University, Pittsburgh, PA, 1986. H.J.Aa. Jensen and H. &ren, Chem. Phys., 104 (1986) 229. J. Almlof, USIP 74-29, Stockholm University, 1974. P.E.M. Siegbahn, Chem. Phys. Lett., 109 (1984) 417. M. Pelissier, private communication. J.J. Dongarra, Technical Memorandum No. 23, Argonne National Laboratory, Argonne, IL, 1988. H.J.Aa. Jensen and P. Jergensen, J. Chem. Phys., 80 (1984) 1204. A. Sanchez de Me&, Ph.D. Thesis, Ulniversidad de Valencia, Valencia, 1989. H.J.Aa. Jensen, P. Jergensen and H. Agren, J. Chem. Phys., 87 (1987) 451. C.F. Bender andE.R. Davidson, J. Chem. Phys., 47 (1967) 4972.D. Feller andE.R. Davidson, J. Chem. Phys., 74 (1981) 3977. P. Fantucci, V. Bonacic-Koutecky and J. Koutecky, J. Comput. Chem., 6 (1985) 462.