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Dipole moment functions of OH by ab initio effective valence shell hamiltonian method
Volume 150, number 6
CHEMICAL PHYSICS LETTERS
30 September 1988
DIPOLE MOMENT FUNCTIONS OF OH BY AB INITIO EFFECTIVE VALENCE SHELL HAMILTONIAN METHOD Hosung SUN Department of Chemistry, Pusan National University,Pusan 609- 735, Korea
Yoon Sup LEE Department of Chemistry, Korea Advanced Institute of Science and Technology, Seoul 136-791, Korea
and Earl F. FREED The James Franck Instituteand Department of Chemistry, The Universityof Chicago, Chicago, IL 60637, USA Received 16 May 1988
The ab initio effective valence shell Hamiltonian method, which is based upon quasidegenerate many-body perturbation theory, has been extended to calculate molecular properties. This new method is applied to the study of dipole moment functions of the OH molecule and its ions. The calculated results are in good agreement with those from other approaches. The present calculations demonstrate that the effective valence shell Hamiltonian formalism is also a reliable ab initio method for molecular properties.
1. Introduction The effective valence shell Hamiltonian (H’) method is based upon quasidegenerate many-body perturbation theory (QDMBPT) and is an ab initio technique for treating molecular electronic correlation. The method generates an H” that spans a prechosen multidimensional valence space [ l-51. This H’ approach has been shown to provide an accurate representation of valence state energy levels for a number of atomic and molecular systems [6-lo]. An unusual feature of the H” method is that a single H” computation simultaneously produces the energy levels of all valence states of the neutral molecule and its ions. In addition, the H’ method can provide insights into the principles underlying semi-empirical eiectronic structure theories [ 6,7]. Whereas single-reference state many-body perturbation theory (MBPT) has been widely used to describe correlation effects on molecular properties [ 1l-l 41, it is only recently that we have generalized the multireference state H’ method to calculate mo-
lecular properties other than energies [ 15,161. The fact that the H’ method is basically a multireference MBPT makes it of great interest to develop the H’ method for calculating molecular properties and to compare its predictions with other accurate ab initio methods. An understanding of the capabilities of the new H’ formalism for molecular properties requires the consideration of a wide range of H’ computations. ‘However, the only existing application of the H” method for molecular properties is for the dipole and transition dipole moments of the CH molecule at its equilibrium geometry [ 16 1. Here we calculate several dipole moment functions for the OH molecule because these may be tested against available high-quality ab initio theoretical studies [ 17-341. Section 2 briefly shows how to obtain the effective operator A” for a property A, while section 3 describes how the formulation is applied for dipole moments. The dipole moment functions are computed for several states of the neutral OH molecule and the ground states of its positive and negative ions.
0 009-2614/G/% 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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Volume150,number6
CHEMICAL PHYSICSLETTERS
Comparisons are made with previous calculations in section 4.
2. Theory The molecular electronic Hamiltonian H is decomposed into a zeroth-order part Ho and a perturbation V, HE&-l-Y.
(1)
When Ho is taken to be a sum of one-electron Fock operators, the perturbation I/ generates the “correlation” correction. The full many-electron Hilbert space is divided into a primary space with a projection operator POand its orthogonal complement with a projection operator Qo= 1-PO. The POspace contains a set of many-electron basis functions that are “quasidegenerate” with respect to the zeroth-order Hamiltonian Ho, Here POis defined to span the valence space of all distinct configuration state functions involving a filled core and the remaining electrons distributed among the valence orbitals. Hence, the Q. space contains all other configuration state functions with at least one core hole and/or one occupied excited orbital. The full Schrijdinger equation Hy=Ey/
H’y’ = Ey’
(3)
for the projection ~7 = Powi, where the energies E in (3) are the exact eigenvalues of eq. (2) and where H” is called the effective valence shell Hamiltonian. Quasidegenerate perturbation theory gives H’ in lowest non-trivia! order as [ 1] H’=P,,HP,
c
&I
(
where h.c. designates the Hermitian conjugate of the preceding term and PO(A) refers to the projector onto the valence space basis function 1A ) , The sum over states in (4) can be reorganized into 530
expressions involving matrix elements of the one and two-electron parts of Ho and Vwithin the orbital basis. (Final expression for the individual matrix elements of H” in the valence orbital basis are presented in ref. [ 31. ) Perhaps the most remarkable aspect of the H” method is the fact that when the second-order quantized form of H’ is used with a reasonable set of orbitals for the ground state of a molecular system, the eigenvalues of eq. (3) provide an accurate approximation to all valence states of the molecular system and its ions simultaneously without the need for recalculating a new H’ with orbitals optimized individually for these other states [6-l 01. Now consider an operator A whose matrix elements between the normalized full space wavefunctions are desired. The matrix element ( vt 1A 1w) may be transformed with QDMBPT into the matrix elements of an effective valence shell operator A’ between the orthonormal valence space eigenfunctions 1~71 as (~ilAlyivi)=(~~lAl’vJ’li’.
(5)
The specification that A’ be Hermitian and independent of the state vi leads to the lowest non-trivial order perturbation expansion [ 4,5,15,16] A’= PoAPo
(2)
is formally transformed into the PO (valence) space to produce the effective valence shell Schriidinger equation,
+f
30September1988
Thus, we obtain desired expectation values by first solving eq. (3) and then by evaluating the corresponding matrix elements on the right-hand side of eq. (5) using eq. (6). Once this A’ is available, it provides, in principle, all diagonal and off-diagonal matrix elements of A in those vi whose images ~7 lie in the POspace. One object of the present calculation is to determine whether the bond length dependence of these matrix elements can be evaluated accurately from a perturbative truncation such as in eq. (6 ) . Either a direct algebraic method or a many-body diagramatic technique may be employed to reduce eq. (6) to expression for the matrix element of A’ in the valence orbital basis. Formulas for these A’ matrix elements are given in ref. [ 16 1, The resulting equations can be formally written in terms of the core, one-, two-,.., body valence shell operators A:, A;, A;, .... respectively, as
CHEMICALPHYSICSLETTERS
Volume 150, number 6
A’=A;+
CA:+4 I
CA;+...,
i#j
(7)
where A: is the constant contribution from the core, A: is a one-electron operator with matrix elements ( u ]A: 1v’) in the valence orbital basis { v},etc. The original full operator A may have one- and two-electron contributions, but our computations consider only the one-electron dipole operator for which the lowest-order expansion in eq. (6) generates core, oneand two-electron effective operators. The emergence of the non-classical A; in this case parallels that of three-electron operator contributions to the leading non-trivial order H’ of (4). The perturbation procedure is completely specified once the orbital basis and Ho are chosen. Diagonalization of the perturbative H” in the POspace yields the valence state energies and eigenfunctions {VT}.The latter may be employed along with eq. (5) to compute expectation values of any operator A by use of the effective valence shell operator A’.
3. Calculations The dipole operator P in atomic unit is defined as (8) where 2, is the charge on the nucleus LYat the position R,, and r is a sum of individual electron position operators {ri} multiplied by the electron charge ( - l), i.e. r= - Ciri (in au). Within the BornOppenheimer approximation, the nuclear contribution IR,Z, is a constant for a given geometry, and it is only necessary to evaluate the diagonal and offdiagonal matrix elements of the electronic position operator r. In the present H’ calculations, we merely replace A by r. A’ is, therefore, r” and the effective matrix elements ($i I r; 1$j > and (@l&j]rrs I $& > are evaluated with the lowest-order A’ expansion. Here $i is a valence spin orbital for electron 1, etc. The constant contribution r: of core electrons to the dipole moment is also computed. The dipole moment of the valence state i is (vi I p) vi> and is evaluated as (~7 I p” I ~7 ) in the H’ formalism with v/r the eigenfunction of the second-order H” [ 221. The nuclear contribution is equal
30 September 1988
to the internuclear distance R (in au) for the OH molecule when the origin of the coordinates is taken at the oxygen nucleus. Thus, a positive value for the dipole moment indicates partial positive charge on the H atom. The POspace is chosen as the full valence space of OH which consists of the space spanned by the 20, 3a, In; and 40 molecular orbitals. The la orbital is the only core orbital, and all remaining orbitals are excited orbitals. The zeroth-order Hamiltonian Ho is constructed as a diagonal matrix in the molecular orbital basis. The choice of the H,-,is quite flexible so long as it is diagonal and the zeroth-order states satisfy certain quasidegeneracy constraints [ 21. The present calculations generate the molecular orbitals from self-consistent-field (SCF) calculations for the ground X zlI state of neutral OH, but the SCF orbital energies for the OH’+ are used in the diagonal elements of Ho. Furthermore, the valence orbital energies arc arithmetically averaged in order to guarantee the strict degeneracy of the valence space. The dipole moment functions of OH are calculated at various internuclear distances from 1.8 to 3.0 au. A single H’ calculation also provides the dipole moments of OH+ and OH- valence states. While all the valence states can be simultaneously described by H’ and 7, we consider only those lowlying states of OH and the ground states of OH+ and OH- for which there are prior high-quality ab initio computations for comparison. The basis set is Dunning’s 4s3p contracted Gaussian-type orbital set with two d polarization functions on oxygen and 2s plus one p on hydrogen. This basis set is selected since extensive configuration interaction (CI) calculations have been reported [ 32,351 with this basis.
4. Results and discussion Table 1 displays a set of matrix elements of the core, one- and two-electron parts of r” in the valence orbital basis as computed at the internuclear distance of 1.85 au. These matrix elements are, of course, repeatedly evaluated at each’ internuclear distance to obtain the dipole moment functions. However, a single set of r” matrix elements is used at one geometry to evaluate expectation values of r” 531
Volume 150, number 6
CHEMICAL PHYSICS LETTERS
30 September 1988
Table 1 Calculated H’ matrix elements (in au) of the effective dipole operator for OH at R= 1.85 au. SCF designates the expectation values of dipole operator between SCF molecular orbitals Matrix element
H’
SCF
Matrix element
H’
r: (2olr;l2o) (3alr;l3a)
-0.000370 -0.264865 -0.118395 -0.040685 - 1.390968 0.489726 0.219238 -0.689990 -0.366822 -0.050382 -0.030326
-0.000306 -0.327365 -0.159425 -0.055278 -1.280121 0.773137 -0.650798 -0.444154 -0.650786 -0.139893 d.082452
( 2020 1ry212020) ( 3a3a 1ry7I3a3a) (40401 rh I404o)
-0.027758 -0.010131 0.228554 -0.015654 0.011757 0.023463 0.046068 -0.001693 -0.026789 0.014258 -0.013643 0.028132 -0.011807 -0.010739 0.004352 -0.007420 0.000102 -0.001006
(tx,Ir;Itrr,)
(Wr;lW (Wr;lls> (3alr;l la> (Wr‘;ll~>
( Wrr,lri~
IWG>
( 3cr2a 1r’;* (3a2a) ( 4a2a 1r;* I4o2a) (4a3ol ry2I4a3a) ( lxX201r;z I lrr,20) (1~~301r;~ Il53a) ( lrr,4alr;, 11x,40) (W4r;2
I $JG
(3a2al rr2 I2o3a) (4o2o( r;* I2o4o) (4o3ol& 13040) ( 17tX201 ri2 12olx,) ( ln,3al ry2I3alrr,) ( 17rX4u~r~,~4017tX) ( l~yl~xlr;2 I 1sW
for all valence states of OH, OH+, and OH-, regardless of the charge and symmetry of the states. As expected, the core part r: = ( lo I r7 I 1o) is quite small, indicating that the polarization of the core orbital by the presence of the H atom is very small. The SCF value of ( lo 1rI 1la) is also given in table 1. The difference between r; and the SCF value represents contributions of core-core correlations to the dipole moment. This difference is tiny, but is a significant portion of the total core contribution, implying that core correlation may also be important for those molecules having significant SCF core contributions. Table 1 also shows matrix elements of the one- and two-electron parts of r” among the valence orbitals. Several of the one-electron matrix elements exhibit significant correlation contributions. Two-electron parts appear in r” but do not exist in SCF calculations for one-electron operators like r. Although the magnitudes of two-electron matrix elements are generally smaller than the one-electron terms, neglect of the two-electron contributions ry2 would lead to completely erroneous values for the dipole moment. Note the rather large size of the 40 matrix elements of r;*. The computed dipole moment functions for the 532
valence states of the neutral OH molecule are given in table 2 along with other ab initio values. Our results are generally in reasonable agreement with the CI calculations. Langhoff et al. only report graphical illustrations of the dipole moment functions for the A %+, B 2C+ and C *E+ states [ 231, and the pattern of dipole moment functions from the H' calculations is very similar to that in ref. [ 231. The small difference may be due to the smaller basis set adopted in the present work. The position of the maximum of the dipole moment function for the X 211state from the H' method differs slightly from other calculations, probably because of our use of the lowest-order A' expansion. The dipole moment functions for the OH+ and OH- ground states are listed in table 3. Our H' values again are in good agreement with ab initio SCEP values. The H' calculations also generate dipole moment functions for the excited states of these ions. It is noted once again that one set of r” matrix elements at a given geometry yields the dipole moments for all states of the neutral OH and the ions OH+ and OH-, and this feature of the method greatly reduce the computational effort. Sources of errors in the current W calculations and possible remedies are as follows: Firstly, the incom-
30 September 1988
CHEMICAL PHYSICS LETTERS
Volume 150, number 6
Table 2 Dipole moments (in au) of the low-lying valence state of OH. R is the internuclear distance in au
R
1.80 1.85 1.90 2.20 2.30 2.40 2.50 2.60 3.00
A%+
X9 H'
CI”’
CI b’
MCSCF SCEP ‘)
Cl”’
H'
CI d’
0.6726 0.6763 0.6792 0.6829 0.6784 0.6705 0.6602 0.6486 0.5564
0.6561 0.6617 0.6665 0.6840 0.6874 0.6822
0.6427 0.648 1 0.6530 0.6725 0.6740 0.6724
0.6410
0.6396 0.644 1 0.6476 0.6546
0.6969 0.7334 0.7693 0.9417 0.9744 0.9927 0.9957 0.9823 0.7759
0.6469 0.6803 0.7133 0.8816
0.6590 0.5848
0.6507 0.6676 0.6681 0.6652 0.6590 0.5670
0.6317 0.5187
0.9535 0.7784
*) Full CI calculation using the same basis set as ours (ref. [ 321). b)CIwithlargerbasisset(ref. [32]). @Ref. [29]. *‘Ref. [ZS]. e, Graphical presentation of dipole moment functions for A zZ +,BZZ+,CzE+statesisprovidedinref.
pleteness of the basis set, which is inherent in any basis set expansion, can be reduced by using large basis sets. Secondly, the third-order H' may be employed in the future to provide a more accurate representation of the {~7). Lastly, we use the non-trivial lowest-order A” expansion of eq. (6), and the evaluation of higher-order A ”expansions is currently under study. In summary, the extension of the effective valence shell Hamiltonian method for properties is explicitly applied to the dipole moment functions of OH. Our computed dipole moment functions are in good agreement with high-quality CI and SCEP calcula-
B*E+
CzE+”
H'
H
-0.1960 -0.3170 -0.4462 -0.8620 -0.8144 -0.7290 -0.6285 -0.5229 -0.1188
0.6977 0.5363 0.2952 -0.3753 -0.2671 -0.1078 0.074 1 0.2667 1.0998
[23].
tions. This comparison supports the utility of the effective Hamiltonian method for molecular properties. The correlated effective valence shell dipole operator is represented in terms of its matrix elements within valence orbital basis, and the properties of these matrix elements are analyzed. This effective operator contains two-electron components in addition to the usual one-electron dipole operators even in the lowest non-trivial order approximations to H' and A’ studied here. The two-electron matrix elements are generally small, but significant enough to assure that semi-empirical electronic structure methods with only one-electron dipole operators must ef-
Table 3 Expectation values in au of dipole operator for the ground state of the OH+ and OH- ions. Origin of coordinates is on the oxygen atom. R is the internuclear distance in au R
1.80 1.85 1.90 2.20 2.30 2.40 2.50 2.60 3.00
X’Z- (OH+)
X ‘E+ (OH-)
H'
MCSCF SCEP ‘)
H'
0.9399 0.9626 0.9865 1.1611 1.2230 1.2845 1.3456 I .4065 1.6450
0.9275
0.3884 0.3711 0.3499 0.1540 0.0855 0.0201 -0.0430 -0.1040 -0.3100
0.9801 1.1484
1.3271 1.6268
MCSCF SCEP a) 0.3222 0.2884 0.1602
-0.0001 -0.2699
“‘Ref. [29].
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Volume 150, number 6
CHEMICAL PHYSICS LETTERS
fectivelybe averagingtwo-electroncontributions into their semi-empirical procedure [ 71.
30 September 1988
[ 141 S.A. Kucharski, Y.S. Lee, G.D. Purvis III and R.J. Bartlett, Phys. Rev. A 29 ( 1984) 1619, and references therein. [ 151 V. Hurtubise and K.F. Freed, to be published.
[ 161 H. Sun and K.F. Freed, J. Chem. Phys. 88 (1988) 2659. 1171 P.E. Cadeand W.M. Huo, J. Chem. Phys.47 (1967) 614.
Acknowledgement
The research is supported by a grant from the Korea Science and Engineering Foundation and by Grant CHE 86-17480 from the National Science Foundation.
References [ 1 ] M.G. Sheppard, K.F. Freed, M.F. Herman and D.L. Yeager, Chem. Phys. Letters 61 (1979) 577. [ 21 M.G. Sheppard and K.F. Freed, J. Chem. Phys. 75 ( 1981) 4507. [ 31 M.G. Sheppard and K.F. Freed, Chem. Phys. Letters 82 (1981) 235. 141 B.H. Brandow, Advan. Quantum Chem. 10 (1977) 188, and references therein. [ 51I. Lindgren, Rept. Progr. Phys. 47 ( 1984) 345, and references therein. [ 6 ] K.F. Freed, in: Modem theoretical chemistry, Vol. 7, ed. G.A. Segal (Plenum Press, New York, 1977). [7] K.F. Freed, Accounts Chem. Res. 16 (1983) 137, and references therein. [8] T. Takadaand K.F. Freed, J. Chem. Phys. 80 (1984) 3253, 3696. [9] H. Sun and K.F. Freed, J. Chem. Phys. 80 (1984) 779. [ LO]X.C. Wang and KF. Freed, J. Chem. Phys. 86 (1987) 2899. [ 111 H.P. Kelly, Advan. Chem. Phys. 14 (1969) 129. [ 121 H.P. Kelly, Phys. Rev. 152 ( 1966) 62. [ 131 B.J. Bartlett and G.D. Purvis III, Phys. Rev. A 20 (1981) 1313.
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[ 181J.O. Arnold, E.E. Whiting and L.F. Scharbaugh, J. Chem. Phys. 64 ( 1976) 325 1. [ 191 R.J. Fallon, I. Tobias and J.T. Vanderslice, J. Chem. Phys. 34 (1961) 167. 1201C.F. Bender and E.R. Davidson, Phys. Rev. 183 (1969) 23. [ 211C.W. Bauschlicher Jr. and S.R. Langhoff, J. Chem. Phys. 87 (1987) 4665. [ 22] H. Sun, M.G. Sheppard and K.F. Freed, J. Chem. Phys. 74 (1981) 6842. [23] S.R. Langhoff, E.F. van Dishoeck, R. Wetmore and A. Dalgamo, J. Chem. Phys. 77 (1982) 1379. [24] R. De Vivie, C.M. Marian and SD. Peyerimhoff, Chem. Phys. 112 (1987) 347. [25] J.S. Wright, V. Barclay and E. Kruus, Chem. Phys. Letters 122 (1985) 214. [26] J.J. Novoa and F. Mota, Chem. Phys. Letters 123 (1986) 3999. [27] J.V. Ortiz, Chem. Phys. Letters 136 (1987) 387. 1281 S. Chu, M. Yoshimine and B. Liu, J. Chem. Phys. 61 ( 1974) 5389. [29] H.-J. Werner, P. Rosmus and E.-A. Reinsch, J. Chem. Phys. 79 ( 1983) 905. [30] W.J. Stevens, G. Das, AC. Wahl, M. Krauss and D. Neumann, J. Chem. Phys. 61 (1974) 3686. 1311 W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. [ 321S.R. Langhoff, C.W. Bauschlicher Jr. and P.R. Taylor, J. Chem. Phys. 86 (1987) 6992. [33] H.-J. Werner, P. Rosmus, W. SchPtzl and W. Meyer, J. Chem. Phys. 80 (1984) 831. [ 341 D.J. Rodgersand P.J. Sane, Chem. Phys. Letters 143 (1988) 235. [ 351 T.H. Dunning Jr., J. Chem. Phys. 53 (1970) 2823.
CHEMICAL PHYSICS LETTERS
30 September 1988
DIPOLE MOMENT FUNCTIONS OF OH BY AB INITIO EFFECTIVE VALENCE SHELL HAMILTONIAN METHOD Hosung SUN Department of Chemistry, Pusan National University,Pusan 609- 735, Korea
Yoon Sup LEE Department of Chemistry, Korea Advanced Institute of Science and Technology, Seoul 136-791, Korea
and Earl F. FREED The James Franck Instituteand Department of Chemistry, The Universityof Chicago, Chicago, IL 60637, USA Received 16 May 1988
The ab initio effective valence shell Hamiltonian method, which is based upon quasidegenerate many-body perturbation theory, has been extended to calculate molecular properties. This new method is applied to the study of dipole moment functions of the OH molecule and its ions. The calculated results are in good agreement with those from other approaches. The present calculations demonstrate that the effective valence shell Hamiltonian formalism is also a reliable ab initio method for molecular properties.
1. Introduction The effective valence shell Hamiltonian (H’) method is based upon quasidegenerate many-body perturbation theory (QDMBPT) and is an ab initio technique for treating molecular electronic correlation. The method generates an H” that spans a prechosen multidimensional valence space [ l-51. This H’ approach has been shown to provide an accurate representation of valence state energy levels for a number of atomic and molecular systems [6-lo]. An unusual feature of the H” method is that a single H” computation simultaneously produces the energy levels of all valence states of the neutral molecule and its ions. In addition, the H’ method can provide insights into the principles underlying semi-empirical eiectronic structure theories [ 6,7]. Whereas single-reference state many-body perturbation theory (MBPT) has been widely used to describe correlation effects on molecular properties [ 1l-l 41, it is only recently that we have generalized the multireference state H’ method to calculate mo-
lecular properties other than energies [ 15,161. The fact that the H’ method is basically a multireference MBPT makes it of great interest to develop the H’ method for calculating molecular properties and to compare its predictions with other accurate ab initio methods. An understanding of the capabilities of the new H’ formalism for molecular properties requires the consideration of a wide range of H’ computations. ‘However, the only existing application of the H” method for molecular properties is for the dipole and transition dipole moments of the CH molecule at its equilibrium geometry [ 16 1. Here we calculate several dipole moment functions for the OH molecule because these may be tested against available high-quality ab initio theoretical studies [ 17-341. Section 2 briefly shows how to obtain the effective operator A” for a property A, while section 3 describes how the formulation is applied for dipole moments. The dipole moment functions are computed for several states of the neutral OH molecule and the ground states of its positive and negative ions.
0 009-2614/G/% 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
529
Volume150,number6
CHEMICAL PHYSICSLETTERS
Comparisons are made with previous calculations in section 4.
2. Theory The molecular electronic Hamiltonian H is decomposed into a zeroth-order part Ho and a perturbation V, HE&-l-Y.
(1)
When Ho is taken to be a sum of one-electron Fock operators, the perturbation I/ generates the “correlation” correction. The full many-electron Hilbert space is divided into a primary space with a projection operator POand its orthogonal complement with a projection operator Qo= 1-PO. The POspace contains a set of many-electron basis functions that are “quasidegenerate” with respect to the zeroth-order Hamiltonian Ho, Here POis defined to span the valence space of all distinct configuration state functions involving a filled core and the remaining electrons distributed among the valence orbitals. Hence, the Q. space contains all other configuration state functions with at least one core hole and/or one occupied excited orbital. The full Schrijdinger equation Hy=Ey/
H’y’ = Ey’
(3)
for the projection ~7 = Powi, where the energies E in (3) are the exact eigenvalues of eq. (2) and where H” is called the effective valence shell Hamiltonian. Quasidegenerate perturbation theory gives H’ in lowest non-trivia! order as [ 1] H’=P,,HP,
c
&I
(
where h.c. designates the Hermitian conjugate of the preceding term and PO(A) refers to the projector onto the valence space basis function 1A ) , The sum over states in (4) can be reorganized into 530
expressions involving matrix elements of the one and two-electron parts of Ho and Vwithin the orbital basis. (Final expression for the individual matrix elements of H” in the valence orbital basis are presented in ref. [ 31. ) Perhaps the most remarkable aspect of the H” method is the fact that when the second-order quantized form of H’ is used with a reasonable set of orbitals for the ground state of a molecular system, the eigenvalues of eq. (3) provide an accurate approximation to all valence states of the molecular system and its ions simultaneously without the need for recalculating a new H’ with orbitals optimized individually for these other states [6-l 01. Now consider an operator A whose matrix elements between the normalized full space wavefunctions are desired. The matrix element ( vt 1A 1w) may be transformed with QDMBPT into the matrix elements of an effective valence shell operator A’ between the orthonormal valence space eigenfunctions 1~71 as (~ilAlyivi)=(~~lAl’vJ’li’.
(5)
The specification that A’ be Hermitian and independent of the state vi leads to the lowest non-trivial order perturbation expansion [ 4,5,15,16] A’= PoAPo
(2)
is formally transformed into the PO (valence) space to produce the effective valence shell Schriidinger equation,
+f
30September1988
Thus, we obtain desired expectation values by first solving eq. (3) and then by evaluating the corresponding matrix elements on the right-hand side of eq. (5) using eq. (6). Once this A’ is available, it provides, in principle, all diagonal and off-diagonal matrix elements of A in those vi whose images ~7 lie in the POspace. One object of the present calculation is to determine whether the bond length dependence of these matrix elements can be evaluated accurately from a perturbative truncation such as in eq. (6 ) . Either a direct algebraic method or a many-body diagramatic technique may be employed to reduce eq. (6) to expression for the matrix element of A’ in the valence orbital basis. Formulas for these A’ matrix elements are given in ref. [ 16 1, The resulting equations can be formally written in terms of the core, one-, two-,.., body valence shell operators A:, A;, A;, .... respectively, as
CHEMICALPHYSICSLETTERS
Volume 150, number 6
A’=A;+
CA:+4 I
CA;+...,
i#j
(7)
where A: is the constant contribution from the core, A: is a one-electron operator with matrix elements ( u ]A: 1v’) in the valence orbital basis { v},etc. The original full operator A may have one- and two-electron contributions, but our computations consider only the one-electron dipole operator for which the lowest-order expansion in eq. (6) generates core, oneand two-electron effective operators. The emergence of the non-classical A; in this case parallels that of three-electron operator contributions to the leading non-trivial order H’ of (4). The perturbation procedure is completely specified once the orbital basis and Ho are chosen. Diagonalization of the perturbative H” in the POspace yields the valence state energies and eigenfunctions {VT}.The latter may be employed along with eq. (5) to compute expectation values of any operator A by use of the effective valence shell operator A’.
3. Calculations The dipole operator P in atomic unit is defined as (8) where 2, is the charge on the nucleus LYat the position R,, and r is a sum of individual electron position operators {ri} multiplied by the electron charge ( - l), i.e. r= - Ciri (in au). Within the BornOppenheimer approximation, the nuclear contribution IR,Z, is a constant for a given geometry, and it is only necessary to evaluate the diagonal and offdiagonal matrix elements of the electronic position operator r. In the present H’ calculations, we merely replace A by r. A’ is, therefore, r” and the effective matrix elements ($i I r; 1$j > and (@l&j]rrs I $& > are evaluated with the lowest-order A’ expansion. Here $i is a valence spin orbital for electron 1, etc. The constant contribution r: of core electrons to the dipole moment is also computed. The dipole moment of the valence state i is (vi I p) vi> and is evaluated as (~7 I p” I ~7 ) in the H’ formalism with v/r the eigenfunction of the second-order H” [ 221. The nuclear contribution is equal
30 September 1988
to the internuclear distance R (in au) for the OH molecule when the origin of the coordinates is taken at the oxygen nucleus. Thus, a positive value for the dipole moment indicates partial positive charge on the H atom. The POspace is chosen as the full valence space of OH which consists of the space spanned by the 20, 3a, In; and 40 molecular orbitals. The la orbital is the only core orbital, and all remaining orbitals are excited orbitals. The zeroth-order Hamiltonian Ho is constructed as a diagonal matrix in the molecular orbital basis. The choice of the H,-,is quite flexible so long as it is diagonal and the zeroth-order states satisfy certain quasidegeneracy constraints [ 21. The present calculations generate the molecular orbitals from self-consistent-field (SCF) calculations for the ground X zlI state of neutral OH, but the SCF orbital energies for the OH’+ are used in the diagonal elements of Ho. Furthermore, the valence orbital energies arc arithmetically averaged in order to guarantee the strict degeneracy of the valence space. The dipole moment functions of OH are calculated at various internuclear distances from 1.8 to 3.0 au. A single H’ calculation also provides the dipole moments of OH+ and OH- valence states. While all the valence states can be simultaneously described by H’ and 7, we consider only those lowlying states of OH and the ground states of OH+ and OH- for which there are prior high-quality ab initio computations for comparison. The basis set is Dunning’s 4s3p contracted Gaussian-type orbital set with two d polarization functions on oxygen and 2s plus one p on hydrogen. This basis set is selected since extensive configuration interaction (CI) calculations have been reported [ 32,351 with this basis.
4. Results and discussion Table 1 displays a set of matrix elements of the core, one- and two-electron parts of r” in the valence orbital basis as computed at the internuclear distance of 1.85 au. These matrix elements are, of course, repeatedly evaluated at each’ internuclear distance to obtain the dipole moment functions. However, a single set of r” matrix elements is used at one geometry to evaluate expectation values of r” 531
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Table 1 Calculated H’ matrix elements (in au) of the effective dipole operator for OH at R= 1.85 au. SCF designates the expectation values of dipole operator between SCF molecular orbitals Matrix element
H’
SCF
Matrix element
H’
r: (2olr;l2o) (3alr;l3a)
-0.000370 -0.264865 -0.118395 -0.040685 - 1.390968 0.489726 0.219238 -0.689990 -0.366822 -0.050382 -0.030326
-0.000306 -0.327365 -0.159425 -0.055278 -1.280121 0.773137 -0.650798 -0.444154 -0.650786 -0.139893 d.082452
( 2020 1ry212020) ( 3a3a 1ry7I3a3a) (40401 rh I404o)
-0.027758 -0.010131 0.228554 -0.015654 0.011757 0.023463 0.046068 -0.001693 -0.026789 0.014258 -0.013643 0.028132 -0.011807 -0.010739 0.004352 -0.007420 0.000102 -0.001006
(tx,Ir;Itrr,)
(Wr;lW (Wr;lls> (3alr;l la> (Wr‘;ll~>
( Wrr,lri~
IWG>
( 3cr2a 1r’;* (3a2a) ( 4a2a 1r;* I4o2a) (4a3ol ry2I4a3a) ( lxX201r;z I lrr,20) (1~~301r;~ Il53a) ( lrr,4alr;, 11x,40) (W4r;2
I $JG
(3a2al rr2 I2o3a) (4o2o( r;* I2o4o) (4o3ol& 13040) ( 17tX201 ri2 12olx,) ( ln,3al ry2I3alrr,) ( 17rX4u~r~,~4017tX) ( l~yl~xlr;2 I 1sW
for all valence states of OH, OH+, and OH-, regardless of the charge and symmetry of the states. As expected, the core part r: = ( lo I r7 I 1o) is quite small, indicating that the polarization of the core orbital by the presence of the H atom is very small. The SCF value of ( lo 1rI 1la) is also given in table 1. The difference between r; and the SCF value represents contributions of core-core correlations to the dipole moment. This difference is tiny, but is a significant portion of the total core contribution, implying that core correlation may also be important for those molecules having significant SCF core contributions. Table 1 also shows matrix elements of the one- and two-electron parts of r” among the valence orbitals. Several of the one-electron matrix elements exhibit significant correlation contributions. Two-electron parts appear in r” but do not exist in SCF calculations for one-electron operators like r. Although the magnitudes of two-electron matrix elements are generally smaller than the one-electron terms, neglect of the two-electron contributions ry2 would lead to completely erroneous values for the dipole moment. Note the rather large size of the 40 matrix elements of r;*. The computed dipole moment functions for the 532
valence states of the neutral OH molecule are given in table 2 along with other ab initio values. Our results are generally in reasonable agreement with the CI calculations. Langhoff et al. only report graphical illustrations of the dipole moment functions for the A %+, B 2C+ and C *E+ states [ 231, and the pattern of dipole moment functions from the H' calculations is very similar to that in ref. [ 231. The small difference may be due to the smaller basis set adopted in the present work. The position of the maximum of the dipole moment function for the X 211state from the H' method differs slightly from other calculations, probably because of our use of the lowest-order A' expansion. The dipole moment functions for the OH+ and OH- ground states are listed in table 3. Our H' values again are in good agreement with ab initio SCEP values. The H' calculations also generate dipole moment functions for the excited states of these ions. It is noted once again that one set of r” matrix elements at a given geometry yields the dipole moments for all states of the neutral OH and the ions OH+ and OH-, and this feature of the method greatly reduce the computational effort. Sources of errors in the current W calculations and possible remedies are as follows: Firstly, the incom-
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Table 2 Dipole moments (in au) of the low-lying valence state of OH. R is the internuclear distance in au
R
1.80 1.85 1.90 2.20 2.30 2.40 2.50 2.60 3.00
A%+
X9 H'
CI”’
CI b’
MCSCF SCEP ‘)
Cl”’
H'
CI d’
0.6726 0.6763 0.6792 0.6829 0.6784 0.6705 0.6602 0.6486 0.5564
0.6561 0.6617 0.6665 0.6840 0.6874 0.6822
0.6427 0.648 1 0.6530 0.6725 0.6740 0.6724
0.6410
0.6396 0.644 1 0.6476 0.6546
0.6969 0.7334 0.7693 0.9417 0.9744 0.9927 0.9957 0.9823 0.7759
0.6469 0.6803 0.7133 0.8816
0.6590 0.5848
0.6507 0.6676 0.6681 0.6652 0.6590 0.5670
0.6317 0.5187
0.9535 0.7784
*) Full CI calculation using the same basis set as ours (ref. [ 321). b)CIwithlargerbasisset(ref. [32]). @Ref. [29]. *‘Ref. [ZS]. e, Graphical presentation of dipole moment functions for A zZ +,BZZ+,CzE+statesisprovidedinref.
pleteness of the basis set, which is inherent in any basis set expansion, can be reduced by using large basis sets. Secondly, the third-order H' may be employed in the future to provide a more accurate representation of the {~7). Lastly, we use the non-trivial lowest-order A” expansion of eq. (6), and the evaluation of higher-order A ”expansions is currently under study. In summary, the extension of the effective valence shell Hamiltonian method for properties is explicitly applied to the dipole moment functions of OH. Our computed dipole moment functions are in good agreement with high-quality CI and SCEP calcula-
B*E+
CzE+”
H'
H
-0.1960 -0.3170 -0.4462 -0.8620 -0.8144 -0.7290 -0.6285 -0.5229 -0.1188
0.6977 0.5363 0.2952 -0.3753 -0.2671 -0.1078 0.074 1 0.2667 1.0998
[23].
tions. This comparison supports the utility of the effective Hamiltonian method for molecular properties. The correlated effective valence shell dipole operator is represented in terms of its matrix elements within valence orbital basis, and the properties of these matrix elements are analyzed. This effective operator contains two-electron components in addition to the usual one-electron dipole operators even in the lowest non-trivial order approximations to H' and A’ studied here. The two-electron matrix elements are generally small, but significant enough to assure that semi-empirical electronic structure methods with only one-electron dipole operators must ef-
Table 3 Expectation values in au of dipole operator for the ground state of the OH+ and OH- ions. Origin of coordinates is on the oxygen atom. R is the internuclear distance in au R
1.80 1.85 1.90 2.20 2.30 2.40 2.50 2.60 3.00
X’Z- (OH+)
X ‘E+ (OH-)
H'
MCSCF SCEP ‘)
H'
0.9399 0.9626 0.9865 1.1611 1.2230 1.2845 1.3456 I .4065 1.6450
0.9275
0.3884 0.3711 0.3499 0.1540 0.0855 0.0201 -0.0430 -0.1040 -0.3100
0.9801 1.1484
1.3271 1.6268
MCSCF SCEP a) 0.3222 0.2884 0.1602
-0.0001 -0.2699
“‘Ref. [29].
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fectivelybe averagingtwo-electroncontributions into their semi-empirical procedure [ 71.
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[ 141 S.A. Kucharski, Y.S. Lee, G.D. Purvis III and R.J. Bartlett, Phys. Rev. A 29 ( 1984) 1619, and references therein. [ 151 V. Hurtubise and K.F. Freed, to be published.
[ 161 H. Sun and K.F. Freed, J. Chem. Phys. 88 (1988) 2659. 1171 P.E. Cadeand W.M. Huo, J. Chem. Phys.47 (1967) 614.
Acknowledgement
The research is supported by a grant from the Korea Science and Engineering Foundation and by Grant CHE 86-17480 from the National Science Foundation.
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