J . Phys. Chem. 1988, 92, 3061-3063 P2V9r3
= PAP(r3),
PL?)
= P2(P(?)
(17)
Dimensional arguments on (1 5 ) and (16) then immediately yield P ~ P &) ( P )
= A ~ P ~P /Z (~P ) ,@ 2 3 / 2 ( ~ ) = A ~ P
(18)
p 2 = A4p-'t3
(19)
or, equivalently p2
= A3p2,
Here A I ,A2, A,, and A4 are constants. But in the closed-shell Hartree-Fock case, exactly 1 1 (20) P2(j;?) = iP(3 P(r3, A3 =
4
With this value for A,, ( 1 5 ) , (16), and (19) give
V,, = . 1 r A ~ l p ~ / di: ~(r3
(21)
and 1 N ( N - 1) = - ( ~ T A ~ ) ~ / or ' N nA4 = 2-'/3(N- l y 3 2
(22)
Insertion of (22) in (21) gives, finally
V, = 2-'l3(N- 1 ) 2 / 3 1 p 4 / 3 ( di: ?) = 0.7937(N - 1)2131p4/3(7')di:
(23)
3061
TABLE I: Test of Local Formulas for Electron-Electron Repulsion Eneres (au) atom old new (Z) Hartree-Fock" formulab formula' 1.43 0.95 He (2) 1.03 5.95 5.17 Be (4) 4.49 51.3 54.0 52.3 Ne (10) 195.5 198.0 Ar (18) 201.4 1019 1078 987 Kr (36) Xe (54) 2701 2489 2586 7423 7749 Rn (86) 8244
"Values from Clementi, E.; Roetti, C. A t . Data Nucl. Data Tables 1974, 14, 177. *Equation 8 of text and ref 2. CEquation23 of text. This is the desired approximate formula for Vee.6 Numerical Values and Discussion The reduction of (23) to (8) is trivial for large N, there is only a 5% difference in the values of the constant. Equation 8 already being known to give reasonably good numerical prediction^,^ ( 2 3 ) will as well. Some values are given in Table I. Note that (23) is good for H and He, while (8) fails for them. For high N , (23) remains superior. The interest in (23) is not so much in it as a tool for direct numerical prediction, of course, but as a component in densityfunctional-theory models of Thomas-Fermi or X a type.
Electronic-Structure Methods for Heavy-Atom Moleculest Russell M. Pitzer* Department of Chemistry, Ohio State University, Columbus, Ohio 4321 0
and Nicholas W. Winter Lawrence Livermore National Laboratory, Livermore. California 94550 (Received: August 17, 1987)
Methods are derived to simplify and expand the scope of ab initio electronic-structure calculations using relativistic core potentials. The spin-orbit operator obtained at the same level of approximation is expressed in a simpler form to facilitate matrix-element computation. Double-group results are used, when sufficient spatial symmetry is present, both to block the Hamiltonian matrix and to make it real, even though the wave functions are necessarily complex.
Introduction Electronic-structure calculations on molecules containing heavy atoms have proven to be done most effectively using relativistically derived core potentials and spin-orbit Other approaches using perturbation theory and all-electron relativistic methods have so far only been successful for smaller molecule^.^ We have reformulated many of the steps needed to carry out relativistic core-potential and spin-orbit calculations in order to make them more practical for polyatomic molecules. Computer programs have been modified extensively to make use of these methods in molecules of various sizes. The steps treated in this paper are ( 1 ) modifying the form of the spin-orbit operator obtained with relativistic core potentials, (2) using double-group-adapted linear combinations of Slater determinants to block diagonalize the Hamiltonian matrix, and (3) deriving a method, given sufficient spatial symmetry, to make all the Hamiltonian matrix elements real. Further work to develop methods of evaluating atomic-orbital integrals of core potentials This work was supported by the National Science Foundation under Grant CHE-8312286, by the Division of Materials Science of the Office of Basic Energy Sciences, Department of Energy, and by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
0022-3654/88/2092-3061$01.50/0
and spin-orbit operators will be given elsewhereS6 Form of Spin-Orbit Operator Relativistic core potentials (derived from Dirac-Fock atomic calculations) depend on both the I (orbital) and j (total) angular momentum magnitude quantum numbers of the atomic shells they r e p r e ~ e n t .When ~ used, therefore, they must be multiplied by projection operators that operate on both space and spin functions:
The summation limits in this and subsequent summations are determined by standard properties of angular momentum operators: ( I ) Pitzer, K. S . Int. J . Quantum Chem. 1984, 25, 131. (2) Krauss, M.; Stevens, W. J. Annu. Rev. Phys. Chem. 1984, 35, 357. (3) Christiansen, P. A,; Ermler, W. C.; Pitzer, K. S. Annu. Reu. Phys. Chem. 1985, 36, 407. (4) Balasubramanian. K.: Pitzer. K. S . Adu. Chem. Phvs. 1987. 67. 287. (Si Ermler, W. C.; Ross, R. B.; Christiansen, P. A. Adu: Quantum Chem. 1988, 19, 139. ( 6 ) Pitzer, R. M.; Winter, N. W., paper in preparation.
0 1988 American Chemical Society
3062 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 mj:
-j to +j
'/>I
11 -
j:
m,:
TABLE I: D,' Point GlOUD
+ '/2
to I
-I to +I
m,:
-72
to
Pitzer and Winter
B, E, B, E
+'/2
It is considerably simpler to deal with projection operators that only operate on spatial coordinates:
6, = Cllml)(Imll mi
Thus the I- and j-dependent relativistic effective (core) potentials (REPS) are often averaged over j values to obtain I-dependent averaged relativistic effective potentials (AREPs). Note that
Ealj= a, 1
where a unit spin operator has been omitted. This averaging of potentials is not an approximation if the differences of the REPS from the AREPSs are included in the calculation as well. These differences, however, constitute a one-electron spin-orbit operator7,*that provides an approximate description of the true (one- and two-electron) spin-orbit interactions in the same sense that the AREPs provide an approximate description of the true (one- and two-electron) Coulomb's law interactions.
IFo= C [ A V E P ( r ) / ( 2+1 1)1[~6/,/+,/2 - (1 + 1)~/,/-,/*1 I
= q K / , ( r ) - q!Y/2(r) The 1 summation in I?O goes from 1 to a maximum value usually selected as one larger than the largest I value contained in the core of the atom. It is useful to express EiSO in terms of 6,projection operators also.' The simplest way to do so is $-anticipate a form similar to the widell! _used approximate [(r)l.s' expression. The matrix elements of 1 4 between lJ,m, states are, using
I.? = (j2 - 12 - $)/2
- A
1 1 1
-1 -1
-1 1 -1
2 -2
0
0
1 1 1
1
( I , I + ~ , , ~ , ~ . ~ ~ I , I=- o~ , , ~ I, '>) 1
1 0
Y,@r,a"+Pb x,l,,J,,aa-PP (a$)
This_form-of Pois particularly $onve@tnt in that integrals of E/O,t/(r)lxO,,C / O , t / ( r ) ~ y OC/O/&(r)Lo/ /, can be computed and stored over spatial functions rather than integrals of Poover space-spin functions. The integrals of s' over spin functions are easily incorporated at a later stage. Subroutines needed to evaluate the spatial integrals over symmetry orbitals of generally contracted Gaussian orbitals6 have been added to ARGOS (Argonne-Ohio State integral program). Many-Electron Functions Once spin-orbit terms are included in the Hamiltonian, neither spatial symmetry operators nor spin angular momentum operators commute with the Hamiltonian. As is well-known, the symmetry operators which operate on both space and spin coordinates may still commute with the Hamiltonian, and the group of such operators is called the double group.9 The Hamiltonian matrix may then be blocked by choosing the many-electron basis functions to transform according to the irreducible representations of the double group. Since heavy atoms are seldom accurately described by either of the simple coupling schemes, LS or jj, we must expect to include all of the many-electron functions of a given double-group symmetry that arise from an electron configuration. For this reason we make up these functions in the simplest way possible from products of many-electron spatial and spin functions. A simple example involves the use of D,', the D2 double group (Table I). The one-electron spin functions CY,@transform as E , and once the matrix representations for this basis are worked out, it is straightforward to work out the transformation properties of many-electron spin functions; the two-electron spin functions are shown in Table I. Consideration of spin functions for more electrons easily shows their transformation properties to be
odd number of electrons: ( I,I+'/~,~,I~.;~I,I+]/~,MI/) = (1/2)6m1m1
-1 -1
E
even number of electrons: A l + A , Bl
+ + B2, in equal numbers
The only exception to this pattern is for zero electrons-trivially one spin function of A , symmetry. The double-group-adapted functions for an odd number of Recognizing the diagonal matrix elements of 1,; in the projecelectrons are just single Slater determinants and for an even tion-operator part of Posuggests number of electrons are sums and differences of Slater determinants and the corresponding Slater determinants with all the 38/,/+1/2 - ( l + 1)Q,l-l/2 = 2 C [ I ~ , ~ + ~ * , ~ , ) ( I , I + X , ~ , I ~ . ~ I , I + X , ~ , '+ ) ( I , ~ + ' / ~spins , ~Thus / ~reversed. I all of the many-electron functions arising from an electron m/ ml configuration can be characterized by three symmetry labels: l,l-'/2,mJ)(I,l-X,m,lI.sll,l-X,m,') (lJ-h,m,'Il spatial, spin, and overall. Only functions of the same overall symmetry can have nonzero matrix elements of the Hamiltonian between them. Accordingly, Hamiltonian matrices for systems with odd numbers of electrons can be blocked into two subblocks: those with overall symmetry properties the same as those of a(E+) and those with overall symmetry properties the same as those of P(E-). Similarly, Hamiltonian matrices for systems with an even Although onlyg>e of the operators is needed because 6, number of electrons can be blocked into four subblocks, correcommutes with ?.I and is idempotent, it is convenient to keep both sponding to overall symmetry properties of A I ,A,, B , , and B,. of them when deriving integral formulas.6 Thus The C, double group, C,', is isomorphous with D;, while the D2hdouble group, D2hr,is the direct product of D,' with C,, the simple inversion group. Since ARGOS was originally designed to be used with graphical-unitary-group configuration-interaction wher'e programs, for which the use of point-group symmetry higher than b ( r ) = 2 A V e P ( r ) / ( 2 1 1) D Z his difficult,'O we have so far also confined ourselves to con+ A
(l,~-'/,,m,~~.sl~,~-~,,m/l) = -((I
+ 1)/2)6~]~,,I> 1
- I
+
(7) Hafner, P.; Schwarz, W. H. E. Chem. Phys. Lett. 1979, 65, 537. (8) Ermler, W. C.;Lee, Y. S.; Christiansen, P. A.; Pitzer, K. S. Chem. Phys. Lett. 1981, 81, 7 0 .
(9) Herzberg, G. Molecular Spectra and Molecular Structure, Vol. III, Electronic Structure of Polyatomic Molecules; Von Nostrand Reinhold: New York, 1966.
J . Phys. Chem. 1988, 92, 3063-3069
TABLE I11 Phase Factors“ To Obtain Real Hamiltonian Matrices
TABLE I 1 Hamiltonian Matrix.’ B , Overall Symmetry (AiBi) (AiBi) (BIAI) (B2B3)
(B3B2)
(BiAi)
(8283)
1
(BA)
i
ER
so,
ER
SOY SOX
SOX SOY
1 1
ER
so,
sideration only of D2{ and its subgroups. Reality of the Hamiltonian Matrix The most computationally convenient general basis functions ~ functions. are real atomic and molecular orbitals-and the C U , spin In these bases, the matrix elements of are, in general, complex, and therefore the wave functions (configuration-interaction coefficients) are also complex. When the point-group symmetry of the system is C2,, D2, D2h,or higher, however, it is possible to show that the Hamiltonian matrix elements are either pure real or pure imaginary and that a simple redefinition of the manyelectron functions can make the entire matrix real. The resulting computational savings in memory and in diagonalization time are quite important. The integrals of i a n d in the real-orbital basis and a,@basis are either pure real or pure imaginary: A
pure real:
ix,
1
ER
(Spatial symmetry,spin symmetry); ER denotes electron repulsion matrix element; SO, denotes x x spin-orbit matrix element, etc.; pure imaginary matrix elements are boldfaced; all others are real.
pure imaginary:
3063
$2
I,, iy,L,iY
Accordingly, integrals of 12, and ?Jzare pure imaginary, and those of I& are pure real. When the system has D2/ symmetry, the x, y , and z components of the angular momentum operators each transform according to different @educible representations (Table I), so that matrix elements of Pobetween symmetry-adapted functions can contain a nonzero contribution from at most one of the x , y , or z terms in the 1.Z dot product. The matrix elements (IO) Shavitt, I. Chem. Phys. Left. 1979, 63, 421.
-i -1
i
For Di group in the order (spatia1,spinloverall);factors for C,’ and Di{ follow directly. must therefore be either pure real or pure imaginary, as in the example in Table 11, which shows a schematic lower half Hamiltonian matrix for wave functions of B1 overall symmetry and the four possible pairs of (spatial symmetry, spin symmetry). Inspection of Table I1 shows that the simple insertion of a factor of i into the definition of the (BIAl) and (B3B2)many-electron basis functions will make the matrix real everywhere and therefore symmetric instead of (complex) Hermitian, with obvious computational benefits. These simple symmetry and matrix-element considerations thus identify the imaginary configuration-interaction coefficients in advance. A set of factors of f 1, * i are given for all D,’cases in Table 111. The factors are given in coupling constant notation and are not unique. The choices made for E overall symmetry are such that the E+ and E- submatrices will be identical. Summary Several methods of simplifying and therefore extending the range of ab initio approximate relativistic electronic-structure calculations have been derived. Since perturbation theory is not used, they are applicable to molecules containing any atom regardless of position in the periodic table. Some spatial symmetry is needed to use all of the methods, but, for example, all linear molecules have more than enough symmetry. Computer programs using these methods have been tested and are currently being applied, with small configuration-interaction wave functions, to the potential curve crossings and avoided crossings of C u F and to the electronic states of uranocene.
Optimization of Wave Function and Geometry in the Finite Basis Hartree-Fock Method Martin Head-Gordon and John A. Pople* Department of Chemistry, Carnegie- Mellon University, Pittsburgh, Pennsylvania 1521 3 (Received: July 14, 1987; In Final Form: October 14, 1987)
In the finite basis Hartree-Fock (HF) method, the energy is a function of independent wave function and geometric variables, which are variational parameters. A definition of the wave function variables as a set of plane rotation angles is adopted, and partial derivatives of the energy with respect to these variables are obtained, leading to a new method for HF wave function determination. Partial derivatives of the energy with respect to geometric variables, with no assumption of wave function optimality, are also derived. Together these two sets of first derivatives permit the simultaneous optimization of geometry and wave function in the H F method. Examples of the application of this procedure to water and methanol are given.
1. Introduction The finite basis Hartree-Fock (HF) method was introduced by Roothaanl and hall^ for closed-shellmolecules and generalized to oDen-shell svstems by PoDle and N e ~ b e t . It ~ is the standard prodedure of a6 initio molecular orbital (MO) theory and yields an approximate energy and wave function at a specified nuclear geometry. The addition of analytic first derivatives of the H F
energy with respect to nuclear coordinates, which was pioneered by Pulay in his- “force m e t h ~ d ” ,has ~ , ~made the calculation of equilibrium H F geometries for medium-sized molecules a routine
- -
*Author to whom correspondence should be addressed.
0022-3654/88/2092-3063$01.50/0
( 1 ) Roothaan, C. C. J. Rev. Mod. Phys. 1951, 23, 69. (2) Hall, G. G. Proc. R. SOC.London, A 1951, 205, 541. (3) PoDle. J. A,: Nesbet. R. K. J . Chem. Phvs. 1954. 22. i 4 j Puiay’, P. MOLPhys: 1969,17, 197.
.
571.
(5) Pulay, P. In Modern Theorefical Chemistry; Schaefer, H. F., Ed.; Plenum: New York, 1977; Vol. 4, pp 153-185.
0 1988 American Chemical Society
