1780
J . Phys. Chem. 1989, 93, 1780-1784
Electronic Structure of Polyatomic Systems Determined with First-Order Correlation Orbitals. Very Accurate Calculations on 10-Electron Molecules Ludwik A d a m o w i c z
Department of Chemistry, University of Arizona, Tucson, Arizona 85721 (Received: May 9, 1988, In Final Form: August 17, 1988)
A recently proposed method for generating the first-order correlation orbital space1 is employed with large Gaussian basis sets to calculate the electronic correlation energy for small polyatomic molecules with many-body perturbation theory and the coupled cluster method. Different sizes of the correlation orbital spaces are considered, and comparison is made with approximate natural orbitals generated by diagonalization of the virtual-virtual part of the second-order density matrix.
Introduction
Expanding the multielectron wave function in terms of Slater determinants is the most prevalent approximation of quantum chemical calculations on electronic structures. Both configuration coefficients,molecular orbital (MO) coefficients, and the primitive (usually atomic, A O ) basis, which defines the Hilbert space for the problem, can be treated as variables in such expansions. In all practical applications of quantum chemical methods, limitations on the number of determinants and the size of the A 0 Hilbert space have to be imposed due to the limited power of computers. Therefore, making the determinantal expansion of the wave function more effective as well as generating the most efficient orbital expansions has been a long-lasting goal for many workers. A variational calculation that uses all Slater determinants possible within a given orbital Hilbert space is called "full CI"* and at present cannot go beyond systems with a very limited number of electrons and a small number of basis functions. With the perturbation theory approach to the electron correlation, one accounts for all possible excitations from the reference zero-order wave function because of the link-diagram t h e ~ r e m but , ~ what remains unaccountable are linked multiexcitation contributions. However, usually, the fourth-order many-body perturbation theory (MBPT)" and coupled cluster (CC) theory7 at various levels of implementation (CCD, CCSD, CCSDT-1, CCSDT),I8 as well as the multireference configuration interaction (MR-CI) method: reproduce the full C I results quite accurately.I0 Unlike the full CI, the MBPT,CC, and MR-CI are applicable for many systems of interest to chemists. The most serious limitation of this applicability comes from the size of the orbital Hilbert space, which for high levels of theory has to be relatively small. Typically, the ~
orbital Hilbert space for a correlated calculation with a single Slater determinant as a reference function is composed of occupied S C F orbitals and correlation orbitals. Recently, we have proposed two methods to generate the correlation space. The first is based on Bethe-Goldstone partitioning of the electronic correlation and was applied in conjunction with numerical orbitals for diatomic neutral and anionic systems." The second, derived from the Hylleraas second-order functional, generates first-order correlation orbitals (FOCO) and applies to conventional basis sets such as Gaussian or Slater orbitals.' The purpose of the present paper is to demonstrate the application of our Hylleraas functional-based method to very extended Gaussian basis sets for small polyatomic molecules, to examine different levels of orbital contractions, and finally to compare the first-order orbitals with numerical orbitals and approximate natural orbitals ( A N O ) . I 2 Methodology
The methodology for generating first-correlation orbitals has been presented bef0re.l The method uses the second-order Hylleraas functional
to optimize the first-order correlation function, @ ( I ) . For a closed-shell system, the zero-order Hamiltonian is defined as the sum of one-electron Fock operators and, therefore, the sum of the zero-order and the first-order energies equals the Hartree-Fock energy. The first-order function consists of determinants that are constructed by double excitations from occupied orbitals to active correlation orbitals
~~~
( 1 ) Adamowicz, L.; Bartlett, R. J. J . Chem. Phys. 1987, 86,6314. (2) Saxe, P.; Schaefer, H. F., 111; Handy, N. C. Chem. Phys. Lett. 1981, 79, 202. Harrison, R. J.; Handy, N. C. Ibid 1983, 95, 386. Brauschlicher, C. W., Jr.; Langhoff, S. R.; Taylor, P. R.; Partridge, H. Chem. Phys. Lett. 1986, 126, 436. Bauschlicher, C. W., Jr.; Taylor, P. R. J . Chem. Phys. 1986, 85, 2779. Bauschlicher, C. W., Jr.; Langhoff, S. R.; Taylor, P. R.; Handy, N. C.; Knowles, P. J. Ibid. 1986, 85. (3) Brueckner, K. A. Phys. Reu. 1955,97, 1353. Brueckner, K. A,; Eden, R. J.; Fransiz, N. C. Phys. Reu. 1955, 93, 1445. Goldstone, J. Proc. R. SOC. London, Ser. A 1957, 239, 267. (4) Bartlett, R. J.; Silver, D. M. Inr. J . Quantum Chem., Symp. 1974, 8, 271; Chem. Phys. Lett. 1974, 29, 199. Silver, D. M.; Bartlett, R. J. Phys. Rev. A 1976, 13, 1. Silver, D. M.; Wilson, S.; Bartlett, R. J. Phys. Reu. 1977, 16, 477. ( 5 ) Bartlett, R. J. Annu. Reu. Chem. Phys. 1981, 32, 359. (6) Frisch, M. J.; Krishnan, R.; Pople, J. A. Chem. Phys. Lett. 1980, 75, 66. (7) b k , J. J . Chem. Phys. 1969,45,4256; Adu. Chem. Phys. 1969.14, 35. (8) (a) Bartlett, R. J. Annu. Reu. Phys. Chem. 1981, 32, 359. (b) Hoffman, M. R.; Schaefer, H. F., 111 Adu. Quantum Chem. 1986, 18, 207. (9) Shepard, R.; Shavitt, I.; Simons, J. J . Chem. Phys. 1982, 76, 543. Lischka, H.; Shepard, R.; Brown, F. B.; Shavitt, I. Int. J . Quantum Chem., Symp. 1981, 15, 91. Sieghbahn, P. E. M.; Almlof, J.; Heiberg, A,; Roos, B. 0. J . Chem. Phys. 1981, 74, 2384. (10) Bartlett, R. J.; Sekino, H.; Purvis, G. D., I11 Chem. Phys. Lett. 1983, 98, 66. Cole, S. J.; Bartlett, R. J. J . Chem. Phys. 1986, 86,873.
a>b
where the first-order configuration amplitudes are (3)
Subscripts i, j, ... indicate occupied orbitals and superscripts a, b, ... are used for virtual orbitals. The optimization process generates the best possible separation of the space of S C F virtual orbitals to active and nonactive orbital sets, such that the value of the Hylleraas functional determined with active orbitals becomes as negative as possible. The size of the active space is a preselected parameter in the FOCO generation. In practice, its value will depend on how many orbitals can we afford or desire to use in the subsequent MBPT/CC calculation. In the next section, we shall see how the change of this parameter influences the value of the correlation energy. (11) Adamowicz, L.; Bartlett, R. J.; McCullough, E. A. Phys. Rev. Lett. 1985, 54, 426. (12) Karlstrom, G.; Jonssom, B.; Roos, B. 0.;Siegbahn, P. E. M. Theor. Chim. Acta 1978, 48, 59.
0022-3654/89/2093-1780$01.50/00 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 1781
Electronic Structure of Polyatomic Systems Active-nonactive orbital transformation is determined in an iterative process similar to the one so effectively used for MCSCF pr0b1ems.l~ It involves generating the gradient, G , and Hessian, H, matrices for the Hylleraas functional (1) with respect to the transformation parameters. The orbital unitary transformation is represented in an exponential form
U+U = 1 U(R) = exp(R) R = -Rt
Applications and Discussion
(4)
with an antisymmetric matrix R defining the rotation parameters. The gradient and Hessian are usual first and second derivatives of J(*) with respect to R:
The optimal orbital rotation results from the Newton-Raphson equation:
R = -G.H-~
(6)
Because of independent evaluation of the first-order configuration amplitudes and the orbital rotation parameters, the present optimization is a two-step procedure. It is executed alternately until the convergence is reached for the value of the Hylleraas functional. In spite of the similarties between the MCSCF procedure and our Hylleraas functional-based method, there is a significant difference between these two approaches. This is that the orbital optimization in our method can be performed for a large number of active and nonactive correlation orbitals due to a limited number of molecular integrals involved in calculating the gradient and Hessian matrices (only integrals that are needed to calculate the second-order energy), contrary to the MCSCF problem where all two-electron integrals are required. This feature allows for application of the present orbital generation procedure to sizable molecular systems of chemical interest. Notice that the FOCO method results from a formal application of the perturbation theory and, therefore, is conceptually consistent with the MBPT/CC approaches. Alternately, the correlation orbital space can be generated utilizing the second-order density:
D(Z’(1) =
s[@(2)*@(0)
+ @ ( l ) * @ ( l ) + @(O)*@(z)]d2 d3 d N
(7)
This has been proposed by Roos et al.9 In their method, so-called approximate natural orbitals (ANO) are obtained by diagonalizing the second-order density matrix. In our comparative calculations, we have diagonalized only the virtual-virtual part of the this matrix: I>J
mental correlation energy. The correlation numerical orbitals were generated based on the Betheaoldstone partitioning of the total electronic correlation into clusters and by applying the numerical MCSCF procedure” independently to each cluster. The final MBPT/CC calculations were performed with the union of all numerical orbitals. The same strategy has also been applied to study ground and excited states of various diatomic anions.15
C
This limited transformation does not alter occupied orbitals and therefore allows retention of their Hartree-Fock canonical character, necessary for the MBPT method as well as for the present implementation of the CCSD + T(CCSD) method. Active ANOs are selected based on the occupation numbers that emerge as eigenvalues from the diagonalization. Finally, we shall also compare the results of the previous two methods with our recent, most accurate, MBPT/CC results obtained with numerical 0rbita1s.I~ The numerical orbital technique is currently limited to atomic and diatomic systems, and consequently, only results for the F H molecule could be subjected to this comparison. In our numerical orbital calculation for the F H molecule we were able to reproduce almost 99% of the experi(13) Dalgaard, E.;Jmgensen, P. J . Chem. Phys. 1978, 69, 3833. Dalgaard, E. Chem. Phys. Lett. 1979, 65, 559. (14) Adamowicz, L.; Bartlett, R. J. Phys. Reu. A 1988, 37, 1.
FH Molecule. One of the primary reasons for developing the FOCO method is to permit the use of much larger basis sets for correlated calculations than is currently possible with traditional approaches. By contracting virtual orbitals through minimization of the second-order functional, we wish to retain most of the essential correlation effects that would be present in a calculation with the extended basis set. The present study concerns the use of large Gaussian basis sets (in the range of about 150 basis functions) for very accurate determination of the electronic correlation energy for small polyatomic molecules. With the current implementation of higher levels of the MBPT/CC methods, a basis set size exceeding 100 becomes computationally prohibitive for practical use. However, one has to resort to much larger basis sets and use the highest levels of theory in order to account for over 95% of the total correlation energy. We think that our FOCO method allows such accuracy routinely for sizable molecular systems, and we shall present some numerical evidence here to substantiate this claim. We shall also study the FOCO generation for these systems in more detail by examining different levels of contractions for the correlation orbital space. The Gaussian basis set used for the F H molecule consists of 124 orbitals. Thirteen uncontracted s functions and eight uncontracted p functions for the fluorine atom are taken from Duijneveldt.l6 The exponents of five d (6 Cartesian angular components) and two f (10 Cartesian angular components) Gaussians have been chosen as (25.0, 7.5, 2.375, 0.75, 0.25) and (3.75, 1.1125), respectively. One s fluorine Gaussian has been eliminated from the original Duijneveldt set due to a linear dependency with the s components of d functions. The hydrogen basis set consists of 9s, 4p, and 3d functions. The first s-type function is a contract of two primitive Gaussians [0.233 (390.7769) + 0.252 (150.999 502); the exponent is in parentheses], and the remaining eight are uncontracted functions with exponents: 58.3094, 22.5239, 8.7006, 3.3609, 1.2983, 0.5015, 0.1937, and 0.07483. The p and d functions are uncontracted Gaussians with exponents (10.0, 3.92857, 0.93041, 0.2877) and (0.2, 1.0, 4.0), respectively. The experimental equilibrium geometry is used in the present calculation, with fluorine-hydrogen separation of 1.7328 au. The present S C F energy of F H (-100.070251 au) is only 0.6 mhartree above the accurate numerical H F result (-100.070 81 au).” The S C F calculation is followed by a partial four-index transformation of two-electron integrals to second-order exchange integrals, which is performed with a three-step algorithm proposed by Saunders and van Lenthe.I8 The generation of the secondorder energy becomes a trivial calculation once integrals are transformed. In Table I we present a comparison of the second-order pair contributions with the best published result obtained with numerical orbitals14 to demonstrate that our Gaussian basis set calculation shows uniformly good agreement with the most accurate energies oscillating around 95%. In the next step, the virtual-virtual part of the second-order density matrix is constructed and diagonalized yielding the ANOs. Four different sizes of the A N 0 correlation space are selected for consideration: they are 16, 31, 43, and 62. This choice is arbitrary, and its purpose is to test the performance of the present method for different numbers of active correlation orbitals. The iterative optimization of the FOCO spaces, of the same sizes and (15) Adamowicz, L.;Bartlett, R. J. J . Chem. Phys. 1985, 83, 6268. (16) Van Duijneveldt, F. B. IBM Res. Rep. 1971, RJ 945. (17) Adamowicz, L.; Bartlett, R. J. J . Chem. Phys. 1986, 84, 6837. (18) Saunders, V. R.;van Lenthe, J. H. Mol. Phys. 1983, 48, 923.
1782 The Journal of Physical Chemistry, Vol. 93, No.5, 1989 TABLE I: FH Molecule; Comparison of the Second-Order Pair Energies Calculated with 124 Gaussian Orbitals with the Previous Calculation bine 127 Numerical OrbitalsI4 (in au) electron pair numerical orbitals GTO lala -0.038 89 -0.036 79 (94.5%) -0.004 81 (95.8) 1a 2 a -0.005 02 2020 -0.01225 -0.01 1 74 (95.8) la3a -0.005 17 -0.004 88 (94.4) -0.026 I O (94.6) 2a3a -0.027 58 3a3a -0.028 25 -0.027 31 (96.7) -0.012 24 (94.0) lala -0.01302 -0.05202 (95.5) 2a1a -0.054 45 -0.08461 (97.1) 3alx -0.087 11 ITlH -0.096 14 -0.091 76 (95.4) TABLE 11: FH Molecule; Comparison of the Second-Order Correlation Energy Calculated with Different Numbers of ANOs and FOCOs (in au) no. of orbitals AN0 FOCO 16 -0.252 37 -0.273 53 31 -0.309 18 -0.325 37 43 -0.330 23 -0.339 36 62 -0.345 97 -0.349 50 TABLE III: FH Molecule; Comparison of the Second-Order Pair Energies Calculated with 31 AN& and 31 FOCOs (in au) electron pair AN0 FOCO lala -0.01693 -0.032 44 1020 -0.002 48 -0.004 12 2a2a -0.01078 -0.01073 -0.003 11 103a -0.003 54 2030 -0.024 09 -0.023 35 -0.025 83 3a3a -0.024 82 laln -0.008 00 -0.009 15 -0.048 25 2a1x -0.048 33 -0.081 87 3ala -0.081 18 laln -0.087 84 -0.087 70
total
-0.309 18
-0.325 37
same symmetry distributions of orbitals, is initiated with ANOs, which seems to be the most reasonable starting guess. In Table I1 we present the values of the second-order energy calculated with different numbers of ANOs and FOCOs. For the same number of orbitals the FOCO energy is consistently better than the A N 0 value. The difference is more significant for 16 and 3 1 orbitals than it is for 43 and 62. Notice that with 62 FOCOs, which is about half of the initial number of all virtual orbitals, the second-order energy is almost identical (99.2%) with the full space result; and with 31 FOCOs, a quarter of the initial set, we get still over 90% of the full space value. Table I11 allows us to examine which pair second-order energies differ the most in the calculations with 31 ANOs and 31 FOCOs. The obvious conclusion is that ANOs greatly underestimate the inner shell correlation, which results from the fact that these energetically important contributions do not manifest themselves with significant occupancies of appropriate correlation orbitals. The A N 0 selection procedure discriminates against these orbitals and puts more emphasis on correlation orbitals for valence electrons. The FOCO method enables high-level MBPT/CC calculations with larger initial basis sets. Until now, molecular applications of MBPT/CC techniques (expecially with triple-excitation contributions) have been limited to rather low-quality basis sets, due
Adamowicz to their unfavorable dependencies on the number of orbitals. The computation time for the MBPT(4) method asymptotially scales as n4,n4ylrt (n, being the number of S C F occupied orbitals and nv,rtbeing the number of S C F virtual orbitals), while the CCSDT-1 ’s time dependence is proportional to n3d4,,*N,, (Nit, is the number of iterations to reach convergence), and for the CCSDT method it is n3mn5nrt*Nlter. Therefore, if a reduction of half of the number of virtual orbitals can be accomplished,it would potentially reduce the computational effort by a factor of 8 and also significantly lower the memory requirements. Table IV presents the MBPT/CC calculations performed with different numbers of FOCOs. Due to our computer limitations, we are unable to use the full virtual space of 119 orbitals at these high levels of theory. Comparing the results for 62 FOCOs and 43 FOCOs indicates that the fourth-order contribution, similar to the second-order contribution, is rather insensitive to the level of orbital reduction. The most stable component of the fourthorder energy is the double-excitation contribution, which at the 16 FOCO level still retains 7 1% of its value calculated with 62 FOCOs. The third-order energy, however small, is altered drastically by the orbital reduction. The CC results are calculated at the CCSD (coupled-cluster method with single- and double-excitation amplitudess) and CCSD + T(CCSD) levels (the latter include the triple-excitation amplitudes in a linear fashion only in the last C C iteration).8 The CCSD and CCSD + T(CCSD) follow the trend of the secondand fourth-order contributions and change relatively little. A reduction from 62 to 43 FOCOs causes a loss of less than 3% of the correlation energy. This observation indicates that with FOCOS and the CCSD + T(CCSD) method we can generate very accurate solutions of the Schriidinger equation, especially for the ground-state and equilibrium molecular geometry, for which CCSD + T(CCSD) was proven to provide results oirtually identical with the “full CI”.I0 To demonstrate this, we present in Table V a comparison of MBPT/CC results obtained with 62 FOCOs and our recent calculation involving numerical orbitals,14 which is, to our knowledge, the most accurate, rigorous, a b initio treatment of the FH molecule published so far. The present CCSD + T(CCSD) result captures 96% of the value obtained with numerical orbitals. The consistency of the higher order correlation corrections at different levels of the reduction of the orbital space suggests using the E(2)obtained with the full set of virtual orbitals with only the higher orders being obtained at the reduced level. This has been tested before,’ and for all cases considered, the results are usually slightly closer to the experimental values. Although the procedure is not rigorous, it can provide a good estimate for the correlation energy. Numbers of the present calculations are shown in the bottom row of Table V, as I?(*) + h[CCSD + T(CCSD)]. H20and NH, Molecules. The Gaussian basis set used for the H 2 0 calculation includes 162 orbitals distributed as 12s, 8p, 5d (with 6 components), and 2f (with 10 components) for the oxygen atom and 8s, 4p, and 3d for the hydrogens. The exponents for s and p functions for oxygen are taken from DuijneveldtI6 (the 14s/8p basis set), and two Gaussians with the largest exponents are contracted as 0.0460 (105 374.9) + 0.3610 (15 679.24). One s function with exponent 1.406 11 had to be eliminated from the original set due to linear dependencies with s components of d orbitals. For the d and f functions the following exponents are used: (20.0, 6.0, 1.9,0.6,0.2) and (3.0,0.9). The hydrogen basis set remains the same as used previously for the FH molecule.
TABLE IV: FH Molecule; Comparison of MBPT/CC Correlation Correction Obtained with Different Number of FOCOs (in au)O E(4)
no. of FOCOs E()) S 62 +O.OOO 68 -0.002 03 43 -0.00008 -0.001 83 31 -0.000 32 -0.001 62 16 -0.001 75 -0.00090 “S,
D
T (100%) -0.004 08 (100%) -0.009 23 (100%) (90%) -0.00397 (97%) -0.00832 (90%) (80%) -0.003 83 (94%) -0.007 18 (78%) (44%) -0.00288 (71%) -0.00278 (30%)
0
total +0.003 65 (100%) -0.01 1 69 (100%) +0.003 33 (91%) -0.01079 (92%) +0.00292 (80%) -0.009 71 (83%) +0.001 25 (34%) -0.00530 (45%)
CCSD 4 . 3 5 0 69 -0.341 38 -0.327 75 -0.27414
D. T, and Q indicate the single-, double-, triple-, and quadruple-excitation contributions to the fourth-order correlation energy.
CCSD + T(CCSD) -0.359 12 -0.34904 -0.334 38 -0.27667
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 1783
Electronic Structure of Polyatomic Systems
TABLE VII: NH, Molecule; SCF and MBPT/CC Results Obtained in a Basis Set of 161 Gaussian Orbitals (in au)
TABLE V FH Molecule; Comparison of the MBPT/CC Calculations with 62 FOCOs and 92 Numerical Orbitals"
SCF total energy orbital energies 1u 2u 3a l?r
FOCO
numerical orbitals
-100.07025
-100.07081
-26.294 07 -1.600 64 -0.767 91 -0.65005
-26.294 57 -1.600 99 -0.768 25 -0.650 39
-0.349 50 -0.01 169 -0.360 51
-0.364 65 +0.001 50 -0.01 1 90 -0.375 05
-0.35069 -0.359 12 -0.361 88
-0.36461 -0.373 54 -0.376 77
SCF total energy experimental estimation orbital no.
E(2) E(') E(4) E(2) + E(')
+O.OOO 68
+ E(4)
coupled cluster CCSD CCSD + T(CCSD) E(2)+ A[CCSD + T(CCSD)] exptl correlation energy
-0.381," -0.377 i O.OOZb
"Reference 22. *Reference 19. TABLE VI: H 2 0 Molecule; SCF and MBPT/CC Results Obtained in a Basis Set of 162 Gaussian Orbitals (in au)
SCF total energy experimental estimation orbital no. 1 2 3 4 5
-76.067 00 -76.068"
1
MBPT second-order pair energies all virtual orbitals electron uair (156) -0.036 90 -0.003 94 -0.01 1 35 -0.003 38 -0.024 06 -0.022 07 -0.004 84 -0.021 62 -0.033 48 -0.03 100 -0.023 64 -0.299 27
orbital energy -20.563 42 -1.35226 -0.584 57 -0.7 17 44 -0.51025
MBPT second-order pair energies all virtual orbitals electron pair (157) -0.037 14 -0.004 43 -0.012 10 -0.004 97 -0.023 79 -0.023 72 -0.003 85 -0.026 88 -0.039 97 -0.024 04 -0.005 63 -0.025 51 -0.041 21 -0.038 25 -0.024 00 -0.335 50
orbital energy -15.538 44 -1.141 50 -0.628 16 -0.429 32
2 3, 5 4
MBPT
-56.223 43 -56.226"
81 FOCOs -0.03675 -0.003 92 -0.01 1 24 -0.003 36 -0.023 73 -0.021 73 -0.004 82 -0.021 42 -0.033 24 -0.030 68 -0.023 43 -0.296 36 -0.01 170 -0.0 1063 -0.3 18 70
coupled cluster CCSD CCSD + T(CCSD) E(2)+ A[CCSD + T(CCSD)]
-0.309 22 -0.3 1827 -0.321 18
exptl correlation energy"
-0.334 f 0.004
Reference 19. 81 FOCOs -0.036 94 -0.004 40 -0.01200 -0.004 95 -0.023 62 -0.023 59 -0.003 83 -0.026 61 -0.039 72 -0.023 78 -0.005 61 -0.025 38 -0.041 08 -0.038 12 -0.023 87 -0.333 50 -0.003 13 -0.01253 -0.349 15
coupled cluster CCSD CCSD + T(CCSD) E(') + A[CCSD + T(CCSD)]
-0.338 30 -0.347 95 -0.349 95
exptl correlation energy
-0.365 f 0.003
Reference 19. In the calculation for the N H 3 molecule, the Gaussian basis consists of 161 functions with 9s, 7p, 4d, and 2f functions for the nitrogen atom and 8s, 3p, and 2d functions for the hydrogens. Also, in this case Duijneveldt's exponents of the 10s/7p basis set are used. Additionally, one s function with the exponent 1.739 195 is eliminated to avoid linear dependencies with the d functions. The nitrogen p and d exponents are (13.5, 2.9, 0.8,0.2) and (2.0, 0 . 6 ) , respectively. The hydrogen basis set consists of the same
8s orbitals used for H F and HzO, 3p functions (exponents 6.2, 1.3, 0.28), and 2d functions (exponents 2.0, 0.4). The calculations are performed at the experimental equilibrium geometries for both molecules with the following Cartesian coordinates (in atomic units): X
H2O
0 H1,2
NH3
N Hl,2 H3
0
f1.430429 0
f1.534320 0
V
0
0 0
-0.885 840 1.771670
2
0
-1.107 157 0
-0.720040 -0.720040
The results are compiled in Table VI for the H 2 0 molecule and in Table VI1 for the NH3 molecule. In both cases the FOCO space contains roughly half of the initial number of virtual orbitals. With this reduction, the loss of the second-order correlation energy is marginal (0.6% or 2 mhartrees for H 2 0 and 1.O% or 3 mhartrees for NH3). The comparison of the second-order pair contributions for full and reduced virtual spaces indicates the same uniform agreement as observed in the F H case. The reduction of the number of correlation orbitals allows us to perform MBPT/CC calculations at the highest implemented level of theory. First, we calculate the third- and fourth-order correlation corrections. It is interesting that while the fourth-order correction is uniformly negative for all three molecules, FH, HzO, and NH3, the thirdorder energy is small and positive for FH and small and negative for H 2 0 ; and for N H 3 it is also negative, but its value exceeds the fourth-order contribution. The C C results are calculated at the CCSD and CCSD + T(CCSD) levels. The triple-excitation contribution accounts consistently for about 3% of the total correlation energy for all molecules considered. The highest level of the theory applied, the CCSD + T(CCSD) method, provides correlation energies, which for all three molecules agrees better than 95% with the experimental estimates of Pople et aI.I9 One may pose a question:
1784
J . Phys. Chem. 1989, 93. 1784-1793
How would the FOCO method perform in calculations on bond-breaking processes? It is known that the second-order energy diverges for such cases, and so does the second-order Hylleraas functional. However, surprisingly enough, the FOCOs determined for nonequilibrium geometries constitute as good basis set for higher orders MBPT/CC calculations as they do for the equilibrium structure. This was illustrated numerically in our previous paper. I The present paper focuses exclusively on the total energy. It is our future goal to apply the FOCO method to study various molecular properties. There is evidence that the method performs well for properties related to the interaction with the electric field.23 Work is in progress to apply FOCOs to study weak intermolecular interactions. The present calculation have been performed on a SCS-40 computer, at the University of Arizona Computer Center. The Gaussian integrals were calculated with a vectorized version of the MOLECULEprogram by Almlof,20and the MBPT/CC calculations were performed with the PROPAGATOR program system.2’ (19) Pople, J. A.; Binkley, J . S. Mol. Phys. 1975, 29, 599. (20) Almlof, J.; Faegri, K., Jr.; Kussell, K. J. Comput. Chem. 1982, 3, 385. (21) The PROPAGATOR program consists of the molecule integral program of J. Almlof, GRNMG, that does SCF calculations and integral transformation of G.Purvis and MBPT/CC for many-body perturbation theory calculations, written by R. J. Bartlett, G . D. Purvis, Y. Lee, S. J. Cole, and R. Harrison. (22) Bender, C. F.; Davidson, E. R. Phys. Reu. 1969, 183, 23. (23) Adamowicz, L.; Bartlett, R. J.; Sadlej, A. J. J. Chem. Phys. 1988, 88, 5149.
Conclusion Very accurate MBPT and coupled-cluster calculations are presented for three polyatomic 10-electron systems, which provide more than 95% of the electronic correlation energy. It is significant that the results have been obtained with conventional Gaussian basis sets, without resorting to explicitly correlated functions, which some believe to be the only functions capable of reproducing the correlation energy to such accuracy. However, the computer time requirements for the highest levels of the MBPT/CC implementations mandate that the number of active correlation orbitals be reduced to the minimum, thereby allowing the calculations to be performed in a realistic time span. The present FOCO procedure, we believe, provides an efficient contraction scheme for virtual orbitals that have emerged from the S C F calculations. In the current implementation of the FOCO method, after correlation orbitals are generated, we perform a full four-index transformation of the two-electron integrals to initiate the MPBT/CC calculation. It is, however, possible to assume that FOCOs are new Gaussian-contracted orbitals and evaluate molecular integrals directly by using them as basis functions. This approach would be especially beneficial when a large number of S C F virtual orbitals are reduced to a relatively small numbers of FOCOs. Work in this direction is in progress. Furthermore, we plan to extend our FOCO procedure to UHF-type zero-order functions. This will allow us to study open-shell radicals and anions. Also, we shall soon present applications of the FOCO procedure to larger systems with over 50 electrons.
Electron Donor-Acceptor Complexes. 2. Evaluation of the Criteria for Ground-State Stability of Weak bw-aw Complexes Using Semiempirical Energy Surfaces William A. Glauser, Douglas J. Raber,* and Brian Stevens Department of Chemistry, University of South Florida, Tampa, Florida 33620 (Received: May 16, 1988)
Semiempirical (MNDO and AM 1) quantum-mechanical calculationsaugmented by dispersion calculations are used to probe the gas-phase, ground-statebehavior of weak br-ar donor-acceptor complexes in both electron donor-acceptor (EDA) complex and exciplex geometries. Computed binding energies are in good agreement with experimental vapor-phase data. The extent to which these calculationsaccord with the predictions of conformationalstability advanced by the orbital correlation scheme is taken as a qualitative measure of the degree of charge transfer in the ground state. Computations reveal little intrinsic difference between EDA complex versus exciplex ground-state stabilities in the gas phase, where it appears that charge-transfer interactions may sometimes be overshadowed by classical long-range interactions, which are less sensitive to conformational changes. Calculated dipole moments and frontier orbital perturbation energies confirm this behavior. The distinction most likely arises in solution due to the dielectric and cage-compression effects of the solvent, which create suitable conditions for charge transfer to occur. The role of configuration interaction in the supermolecule formalism is evaluated and discussed.
Introduction Investigations into molecular complexation provide a fertile testing ground for theoretical descriptions of condensed matter, solvation, phase transitions, and biological molecular recognition. Excited molecular complexes have been implicated as intermediates in collisional electron and energy transfer processes,’ and it has been suggestedZthat reversible complexation may lower the activation barrier to subsequent irreversible thermal chemical reaction. Molecular complexation is predicated upon a delicate balance between short-range (quantal) and long-range (classical) intermolecular forces. At one extreme are weakly bound van der Waals complexes, characterized by loose, nonspecific association and primary stabilized by long-range dispersion interaction^.^ At the ( I ) Masuhara, H.; Mataga, N . Acc. Chem. Res. 1981, 14, 312. (2) Mulliken, R. S. J. Am. Chem. SOC.1952, 74, 811.
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other extreme are hydrogen-bonded complexes that exhibit relatively strong, specific and highly directional binding of a primarily electrostatic nature.4” At intermediate strengths lie donor-acceptor complexes that receive stabilization from charge-transfer (CT) interactions as well as from electrostatic and dispersion A donor-acceptor complex results from an association between closed-shell molecules with definite stoichiometry (usually 1 :1) in which the concentration exceeds that expected for a van der Waals complex.* The charge-transfer stabilization is ostensibly (3) Margenau, H.; Kestner, N. R. Theory of Intermolecular Forces, 2nd ed.; Pergamon Press: New York, 1971. (4) Morokuma, K. J. Chem. Phys. 1971, 55, 1236. (5) Morokuma, K. Acc. Chem. Res. 1977, 10, 294. (6) Morokuma, K.; Kitaura, K. In Molecular Interactions; Ratajczak, H., Orville-Thomas, W. J., Eds.; Wiley: New York, 1980; p 21. ( 7 ) Lathan, W. A,; Pack, G . R.; Morokuma, K. J . Am. Chem. SOC.1975, 97, 6624.
0 1989 American Chemical Society
