Patrick Coffeyl and Karl J U St. Louis University St. Louis, Missouri 63156
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A Pedagogic Approach to Configuration Interaction
The student a ~ ~ m a c h i nQuantum a chemistw for the first time usually h a s little ;difficulty in understanding the a~nroximations involved in the Hartree-Fock method; since explicit consideration of electron repulsion leads to an intractable problem, the Hartree-Fock method allows each electron to see only an auerage repulsive field due to all the other electrons in the system. That this is an approximation, and that it should by the variation principle therefore lead to an enerw defect is also clear. This energy defect is called the correlation energy and is defined as the difference between the exact nonrelativistic energy of the system and the energy predicted by the Hartree-Fock method. The correlation energy arises from the failure of the average field of the Hartree-Fock method to account for the correlated motion of the electrons, i. e., to keep the electrons apart.
Equations Resulting From Mixing Each of Four Confiaurations With ConBpurati~,,
I 1 2
(14'
Simplast LCAO Appmrimation
Nus* + lss) NUS*- ISHI
++ N ( ~ L w+ 2p.d
N@sn 2s") N(2pSr 2pd
CI Equation on Mixing nits, ( 1 # J 2
& ~ ~ +mC ~ ~I C ~~ I m ) I ~ co1.,(1)142)
is the dominant configuration and $I, $2, figurations, can he represented as
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+ a2q.(1)2.,(2)
~
. . . further con-
and the coefficients varied so as to minimize the energy. Usually the configurations +I, $z, etc., are obtained by replacing orbitals occupied in $0 with unoccupied orbitals. is used, Y will he the exact wave If a complete set of function. A complete set is an infinite set, however, and in practice eqn. (1)must he truncated somewhere. Using the Hz molecule as an example, we shall examine the interaction of the dominant ( l a g ) =configuration with four other configurations: l o z ( 2 ~ ~ and ) ~ the , two (lr.)Z. These configurations, their simplest LCAO approximations, and the form eqn. (1) takes when each is individually mixed with the (10,)~ configuration, are shown in the table. In all cases the spin functions have been factored out, and these are spatial configurations. Qualitative plots of the lo, and la. functions versus 2, the intemuclear axis, are given in Figure 2A. I t happens that in optimizing the coefficients of the table that el, cz, and ca are of opposite sign to co and are smaller than co in absolute magnitude.2 If co is arbitrarily taken as positive, the product colag(l) la,(2) is always positive. If both electrons are near one nucleus (on the same side of the node of the la, function), the product la,(l)la.(2) is positive, and multiplication by cl gives a negative numher. is a sum of the two terms: As can be seen in the table, = co1ag(1)1ag(2) cllo,(l)la,(2). is larger when the electrons are near different nuclei, since laU(1)1ad2)is then negative, and multiplication by cl gives a positive numher. The larger the value of the more probable the situation; the (lo.)Z configuration separates the electrons along the internuclear axis and thus accounts for left-right correlation. ~ the ) ~ two ( l a d 2 configThe arguments for the ( 2 ~ and urations are identical to the preceding. A plot of the la, and 20, functions versus p is given in Figure 2B, and of
+
Methods that transcend the Hartree-Fock method and account for electronic correlation are not so easily explicable in terms of physical models. Using Hz as an example, we shall examine configuration interaction, one of the simplest and most widely used methods, and shall present what we hope is a useful pedagogic approach. Coordinate System For a diatomic molecule, the correlation problem may he most conveniently expressed in terms of the cylindrical coordinates 2, p, and q3, where e is the distance along the internuclear axis, 0 is the distance perpendicular to the axis, and 4 is the angle between a reference plane containing the internuclear axis and a plane containing the internuclear axis and the point in question (Fig. 1). Sepa-
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Figure 1. Illustration of cylindrical coordinates p, molecule A-B.
4, and z for a diatomic
ration of the electrons along the z axis is called "leftright" correlation; separation along the p coordinate due to the tendency of one electron to stay far away from the internuclear axis if the other is close to the axis is "inout" correlation; and the tendency of the electrons to he separated hy a value of 4 near 180" (i. e., to be on opposite sides of the internuclear axis) is "angular correlation." Configuration Interaction The Hartree-Fock method yields the best sinxle-configurational approximation to the exact wave function. A confimration is defined as the minimum numher of determinants needed to define a state. One way to get a better wave function is to use more than one configuration. If $0 252
/ Journal of Chemical Education
+
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'Present address: Department of Chemistry, Vanderhilt University, Nashville, Tenn. 37203. ZTbatel, ez, and CJ are of opposite sign to co may be seen from the secular equation for the interaction of one configuration with $0: c d H o o - E ) + c , H m = 0. Ho, is positive and (HOO- Ei is positive: c. is therefore of opposite sign to co. That co is of greater absolute value is simply a matter of definition. $O is defined as that configuration that predominantly contributes to the ground state. 3Davidson. E. R., and Jones, L. L., J. Chem. Phys., 37, 721 (1962).
+.
Figure 2. Qualitative plots of functions Q versus one of the cylindrical coordinates: A 1ou versus 2: 8. 2og versus p: C . 1 u versus 6.The lop function has been included in each plot for comparison. In each case, when both electrons are on the positive side of the mde af Q (solid circles) or on the negative side of the node ( o p n circles). the praduct Q(1) 0(2) is positive. if they are on opposite sides of the node (triangles) their product is negative.
the l u , and one of the l u , functions versus 6 in Figure 2C. In both cases the additional configurations have a node with respect to the coordinate in question, and 0 will be larger when the electrons are separated by this node. The ( 2 ~ 7 , configuration )~ contributes to in-out correlation, and the two ( l n d 2 configurations to angular correlation. Davidson and Jones3 performed a configuration interaction calculation on Ha using natural orbitals, which are those orbitals that give the fastest convergence of eqn. (1). Using just the five configurations of the symmetry we have considered, they accounted for over 90% of the c o r m lation energy. I t is worth noting that if the 2u, function is approximated as N ( 2 s ~+ 2sB), where 2sA and 2sB are Slater-type orbitals, as was done by McLean and coworkers,' it contributes almost nothing to the calculation. This is because a Slater 2s orbital is nodeless and not orthogonal to a Slater 1s orbital. Optimization of the configuration interaction calculation of the table yields two mots, 0 to which
the (1ug)2predominantly contributes, and 0' to which the (2ug)2predominantly contributes. 0 and 0' are orthogonal, and 0' has a node. This calculation is very similar to a Schmidt orthogonalization-0 is almost the same as the ( l u g ) 2configuration, and 0' is formed as a linear comhination of ( l ~ , and ) ~ ( 2 ~SO~as) to~ he orthogonal to 0. Using Slater orbitals, McLean and coworkers had to use an open shell configuration for the ground state to obtain in-outcorrelation. As we go t o higher configurations, the separation into different kinds of correlation becomes impossible. The l u g functions have two nodes, for example, and would be expected to give both left-right and angular effects. I t is thus not possible t o assign exact values to left-right, inout, and angular correlation contributions, although all three are of comparable magnitude. 'McLean, A. D., Weiss, A,, and Yaahimine, A., Rev. Mod. Phys., 32,211 (1960).
Volume 51, Number 4, April 1974
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