R. 1. Snow ond J. L. Bills Brighom Young University Provo, Utoh 84602
II
A Simple h t r c l + i o n of the SCF-LCAO-MO Method
Roothaan's method (1) is the most important method currently used to obtain wave functions for atoms and molecules. His equations are derived and discussed in several textbooks on elementary quantum chemistry (2). In their recent paper on the application of the self-consistent field-linear combination of atomic orbitals-molecular orbital (SCF-LCAO-MO) method to the Hz molecule, Dewar and Kelemen (3) pointed out the paucity of simple illustrations of the Roothaan approach. In this paper, SCF-LCAO equations for the helium atom are derived and are used to obtain the 1s orhital and the energy of the ground state of helium. Several features of the problem recommend it as an introduction to the SCF-LCAO-MO method. These features are: (1) only one-center integrals are necessary; (2) the iteration procedure is simple hut illustrative of the process for more complex systems; (3) the results are very nearly equal to the best SCF results; and (4) a student can carry out the computation on a desk calculator or program the calculation as an exercise in the use of an electronic computer.
and 12 - h=--V 2 '
2
(4)
rl
then
Substituting eqn. (2) into eqns. (3), (5). and (61,we obtain
SCF-LCAO Equations for Helium
We assume that the wave function for the ground state of the helium atom has the form
where 4fi) is the orbital for electron i and afi) and !Xi) are spin functions for electron i. The problem to he considered is the determination of the orbital 4 which gives the best gmund-state energy. If we assume that
where the basis functions x, are Slater-type atomic orbitals and the c, are constants to he determined, then the SCF-LCAO equations give the solution. These equations for helium are derived as follows: The Hamiltonian operator in atomic units for helium is 1 H OV = --V 2
2
1 - 1-V 2 - 1 - f + -
2
r,
rl>
7.2
Then
where for simplicity the c, are assumed to he real and the sums are over all possible values of the indices of the c,. In eqn. (8)
The matrices which correspond to these matrix elements are h, A, and g. Using eqn. ( I ) , we write eqn. (9) explicitly as
where the integrals in the numerator are over both electrons. If we use minimum energy as the criterion to choose the best 4 in eqn. (I), we must find the minimum of E in eqn. (8) by varying the coefficients c,. To avoid quotients in the differentiation we multiply eqn. (8) by ?: c,c,A,, obtaining
We then differentiate eqn. (11) with respect to c, and find
- 2 ~ 4 - 1 ) ( - ; ~ , ~- :)4(l)d~, J4*(l)dl)d~>
If we define
+
a~
(xc,,c,A,JZ
a + E&C,C,A,
a
= 2ac,&cqhnq
a
+ z&~qgm
(12, ~~~*(1)4*(2)$@(1~2)d~,d~~ The various terms in eqn. (12) are evaluated separately by J4*(1)6l)dr,J@*(2)@iZ)drt using the symmetric character of the matrices h, A, and g. A necessary condition for the minimum is
Therefore 506 / Journal of Chemical Education
The parameter c is called the orbital energy, and using eqns. (8)and (18) we see that Furthermore
=
ZCc&
a
+ Ccncqzgm
Calculation of Matrix Elements for Helium
(16)
Now
For simplicity let us use a normalized orbital containing just two basis functions. Then
The parameters i1and izmay be varied to give the best energy, but the coefficients cl and c2 must be optimized for each set of values of II and Tz. The integrals in the matrix elements can be easily evaluated or found in published tables of integrals (4). The elements of h and S, are
Then the last term in eqn. (16) is
Collecting terms from eqns. (13)-(16) and inserting into eqn. (12) we obtain where z = 2 and This equation can be rearranged to the form Fmm eqn. (10) theg,, are
Equation (17) for t = 1, rn is a system of linear homogeneous equations which allow computation of the c, from ht,, gt,, and At; for allowed values of the parameter r . The matrix elements g,, contain the c,; hence the solution of eqn. (17) must be repeated until the coefficients which are obtained are the same as those used to calculate theg,, (hence the name SCF-LCAO). If we let Fw = + g,,
The integrals in eqn. (20) are given by the formulas
eqn. (17) becomes
or in matrix form
where F is called the Fock matrix and
Equation (19) is the matrix form of Roothaan's equationsl and has a nontrivial solution for C only when IF CAI = 0.
'Although eqns. (17) have the same form as Roothaan's equations, his equations differ in that they include exchange terms. If the general Roothaan's equations are applied to the helium atom, the matrix elements g , include negative t e r n s which arise from the exchange operator. The appendix gives a summary of how eqns. (17) must be modified to include exchange, and shows that both sets of equations give the same energy for the ground state of helium. Volume 52, Number 8, August 1975 / 507
and the normalization condition C,2
Values of the parameters which give the minimum energy for the helium atom using two basis functions are f~ = 1.45 and .t2 = 2.90 (5). The energy is improved only slightly when more than two basis functions are used (6). For the above values of the parameters the two-electron integrals and matrix elements h,, and A,, are
(x,x,Ix,x,) = 0.906250 (x2x2Ix2~2)=
~~
(xLxz/xIx1) = 0.954732 ( X ~ X J X J= X JL181482 (x,x,lx,x*) = lxWl@31 (xIxIIx~xZ)= L2Q6660 h,, = - 1848750 h,, h,,
= =
- 1.883523 - 1.595000
A, = L m A,, = 0.838052 In order to find the SCF orbitals we guess an initial set of coefficients c,. When a single basis function is used, the orbital that gives the best energy is 1s (1.6875) (7). Since 1.6875 is closer to (1 than to fz, we expect cl to be larger than cz. We therefore take cl = 1.00 and cz = 0.00 as our initial values. For these values of the coefficients weohtain g,,= 0.906250 g,, = L181482 A,,
g,,
=
F,,
=
=
0.904091
and h,, F,, = h,,
+ 2c,c2hp2+ cs2= 1
For E = -0.984326 we find
c, = 3.6942~
+
+
cZZ= 1 (3.6942cJ2 2(3.6942)(0.838052)~~~ c2 = 0.219059 e, = 0.809249 In order to ensure normalization of @, more figures are retained in the coefficients than are actually significant. The table gives the results for nine iterations of the procedure illustrated. The energy is seen to converge to a constant value in four iterations. The coefficients are stable after eight iterations. The negative of the orbital energy, - 6 , is a n approximation to the ionization potential, as was shown by K w p mans (8). From eqns. (5) and (18) we see t h a t if 4 is normalized E = 2c - (@Igl4)
The wave function for the helium ion He+ is given approximately by
Then if l i s the ionization energy of helium
I
+ g,, = -0.942500
=
E,,.
- E = (mlhlm) - 2(@lhl4)- ( & I d = =
+ g,, = -0.413518 Fll = hla+ gLl = -0.979432 The output coefficients are found for the lower value of t which satisfies the following equation, called the secular equation (see eqn. (19)).
The lower r w t (corresponding to the ground state) is given bv
- ~/C~F,,A,,- (F,, + FdI2- 4(1 - A,*2)(F,,F22- F d ) 2(1 - An2) Evaluating c using the matrix elements given above, we obtain c = -0.984326. The coefficients CI and cz are obtained from
-(4Ihld - (@lgl@) -(
The SCF value of -6 is 0.918. The experimental ionization energy is 0.904 au. The agreement here is better than usual; errors of several percent are common for predictions of this sort (9). The SCF-LCAO-MO treatment of polyatomic molecules differs from the above treatment of helium in the following ways. 1) The matrix elements g ,
include exchange terms. (As shown in the appendix, an exchange term can also be included in the treatment of helium.) 2) All types of integrals are increased in number and in difficulty of evaluation. For a molecule with rn basis orbitals, the number of integrals is mughly proportional to rn4 (10). Many of the integrals involve orbitals centered on different atoms. These multicenter integrals are more difficult to evaluate than the one-center integrals of helium. In approximate MO treatments, many of the integrals are appmrimatedor neplected (11).
Input Coefficients,' Matrix Elements, and Eigenvalues for Helium Iteration
CI
CJ
'The output mefficients of iteration n are the inpvt coefficients for iteration n
508 / Journal of Chemical Education
Fn
FII
+ 1.
Fw
E
3) The dimension of the determinant in the secular equation (21) equals m, the number of AO's in the basis set. A computer is required for efficient solution of secular equations of higher order than three, although the equation can often he factored with the aid of group theory (121. Either way, m energies are obtained as roots of the equation, and the electrons are assimed in pairs to the orbitals of lowest energy. 4) In symmetric molecules, some of the coefficients are determined by symmetry, and need not be found iteratively (3,
This is the farm of the g,, in Roothaan's equations [see Pilar, refe=nce 12aJl. The negative terms are present because for a general closed-shellsystem theenergy by
12).
In spite of these complexities, the SCF-LCAO-MO method has contributed greatly to our understanding of the electronic structure of molecules. With the new generation of computers and more efficient programs, we can expect to see the method extended to moderately large and complex molecules.
Since in the helium atom there is only one occupied molecular orbital the double sum becomes
Appendix Equation (17) can be transformed to Roothaan's equations for helium as follows
+
z c q ( h r v grq - €Arq)
as used in eqn. ( 5 ) of the text. Consequently, the g,, in eqn. (17) have no negative exchange terms. However, when helium is treated with eqn. (22), the g,,' do have negative exchange terms. Then the values of F,,' = h,, + g,' differ from those of F,,, but the same SCFvalues are obtained for e p , c (lower), and E.
Literature Cited
C c,c,cs(X,(1)Xp(2~~~X.(1)Xi2))
PPS
The last equation is true because
1
~e,,c,c,(X,(~)Xp(2)K1Xq(1)X.(2))= q.p.7
~c ~ pc ' ( x r ( 1 ) x p( 2 ) I ~ ~x' ( 1 ) x q( 2 )
1
*.#.*
i?
which follows from the fact that the sum is over all values of q, p, and s. We may thus write Cc&h,,
where 0.8
+ g',,
- cAJ = 0
(22)
(11 Roothaan. C . C . J.. Reu. Mod.Phys.. 23.6Bi19511. 121 See. for example. la1 Pilsr, F. L.. "Elementary quantum Chrmiatry: McGrsw~ Hill. New York, ,968, p . 378: lbl Lovine, I. N.. "Quantum Chemistry,'' Allyn and Bacon. Inc.. Boston. 1970, p. 416 ic) Daudel, R., Lefebure. R., and M o r m c.. " ~ u a n t u mchemistry." wiiey-lnterscienee. N ~ W~ o r k 19%. , p. 462. 131 Derar. M.J.S.. and Kelemen.l.. J . CHEM. EDUC.4R.494119711. i l l Roolhasn.C.C.J..J. C h r m P h y s . 13, 1465i19611. (51 (a) Clemenfi, E., J. Chem. Phys., 40, ISM 119641: (hl Green, L. C.. Mulder. M. M.. L w i s , M. N.. Woll. J. W.. Phys Re". 91. 757 (19541. (61 Roathaan, C , C. J.. Sachs, L. M., and Weirs, A. W.. Re", Mod. Phyr., 32, 186, 119601. (7) Eyrinp. H.. Walter. J.. and Kimhall. G. E.."Quantum Chemistry." John Wiloy & sons. New York, 1944.p. 106. (81 ~ o o p m a n r .T. A,, ~ h y s k o(uirecht). I, IW (1933). ss cited by BUMD. P G.. Coord Chem. Re", 12.37 (19741. (91 Slatei. J. C.. Mann. J. B.. Wilson. T. M.. and Wood. J. H.. Phrs. R a u . 184. 672 (1969). I101 Murroll. J. N., and H a m t , A. J.. "Semi&empirieal Self-consisfenffteld Molecular Orbital Theory of Moleculer," Wiley-lnbr~cience.New York. 1972. P. 9. (111 Pople. J. A . and Beueridee. D. L., ' A p p m r i m a b Molecular Orhital Theory..' McGraw~Hili.New York. 197% Nicholson, B. J., "Advances in Chemical P h y s ~ ies." vd. 18. ~ ~ d i t o r s~rigazine. : I.. and ice. S. A ] . wiley-lnt~rseisnee,y e w York. 1170.p.249. (121 Cotton. F. A . "Chemical Applieationr of Gmup Theory,'' 2nd Ed.. Wiley-Interacionce. New York, 1971, p. 123.
Volume 52, Number 8, August 1975 / 509
