The strengths of the couplings between pairs of CSFs whose energies cross are
evaluated through the SC rules. CSFs that differ by more than two spin-orbital occupancies
do not couple; the SC rules give vanishing Hamiltonian matrix elements for such pairs.
The Slater-Condon Rules
(i) If | > and | '> are identical, then
< | F + G | > = i < i | f | i > + i>j [< ij | g | ij > - < ij | g | ji > ],
where the sums over i and j run over all spin-orbitals in | >;
(ii) If | > and | '> differ by a single spin-orbital
To form the HK,L matrix, one uses the so-called Slater-Condon rules which express
all non-vanishing determinental matrix elements involving either one- or two- electron
operators (one-electron operators are additive and appear as
F = i f(i);
In general, the optimal variational (or perturbative) wavefunction for any (i.e., the
ground or excited) state will include contributions from spin-and space-symmetry adapted
determinants derived from all possible configurations. For example, although the
One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted NElectron Configuration Functions for Any Operator, the Electronic Hamiltonian in
Particular. The Slater-Condon Rules Provide this Capability
I. CSFs Are Used to E
starts with the maximum Ms function and uses spin angular momentum algebra and
orthogonality to form proper spin eigenfunctions, and then employs point group projection
operators (which require the formation of the Rik representation matrices). Antisymmet
The essence of this analysis involves being able to write each wavefunction as a
combination of determinants each of which involves occupancy of particular spin-orbitals.
Because different spin-orbitals interact differently with, for example, a colliding
-R12R21 ]. Evaluating this sum for each of the three symmetries = A1, A2, and E, one
obtains values of 0, 2, and 0 , respectively. That is, the projection of the each of the
original triplet determinants gives zero except for A2 symmetry. This allows one
Here g is the order of the group (the number of symmetry operations in the group- 6 in this
case) and (R) is the character for the particular symmetry whose component in the
direct product is being calculated.
For the case given above, one finds n(a1) =1,
1. The full N! terms that arise in the N-electron Slater determinants do not have to be
treated explicitly, nor do the N!(N! + 1)/2 Hamiltonian matrix elements among the N! terms
of one Slater determinant and the N! terms of the same or another Slater det
2. Is the energy of another 3P state equal to the above state's energy? Of course, but it may
prove informative to prove this.
Consider the MS =0, M L=1 state whose energy is:
2-1/2<[|p1p0| + |p1p0|]| H |<[|p1p0| + |p1p0|]>2-1/2
=1/2cfw_I2p1 + I2p0 + <2p
*i(r) *j(r') g(r,r') k(r)l(r') drdr'.
The notation < i j | k l> introduced above gives the two-electron integrals for the
g(r,r') operator in the so-called Dirac notation, in which the i and k indices label the spinorbitals that refer to the coordinates
Along "reaction paths", configurations can be connected one-to-one according to their
symmetries and energies. This is another part of the Woodward-Hoffmann rules
I. Concepts of Configuration and State Energies
A. Plots of CSF Energies Give Con
of the electrons (the contribution to the dipole operator arising from the nuclear charges - a
Zae2 Ra does not contribute because, when placed between 1 and 2 , this zero-electron
operator yields a vanishing integral because 1 and 2 are orthogonal).
< |1s1s2p02p0| H |1s1s2p-12p1| > = < 2p02p0 | 2p-12p1 >
< |1s1s2p12p-1| H |1s1s2p-12p1| > = < 2p12p-1 | 2p-12p1 >.
Certain of these integrals can be recast in terms of cartesian integrals for which
equivalences are easier to identify as follows:
In all of the above examples, the SC rules were used to reduce matrix elements of
one- or two- electron operators between determinental functions to one- or two- electron
integrals over the orbitals which appear in the determinants. In any ab initio elect
1/2 cfw_< 2s 2s | x x > + < 2s 2s | y y > +i < 2s 2s | y x > -i < 2s 2s | x y > =
< 2s 2s | x x > = K2s,x
(here the two imaginary terms cancel and the two remaining real integrals are equal);
< 2s 2s 2p 0 2p0 > = < 2s 2s | z z > = < 2s 2s | x x > = K2s,x
Kij = <ij | e2/r12 |ji>
are the orbital-level one-electron, coulomb, and exchange integrals, respectively.
Coulomb integrals Jij describe the coulombic interaction of one charge density ( i2
above) with another charge density (j2 above); exchange inte
9. What is the electric dipole matrix elements between the
1 = | 1 1| state and the 1 = 2-1/2[| 1 -1| +| -1 1|] state?
<2-1/2[| 1 -1| +| -1 1|] |r| 1 1|>
= 2-1/2[< -1|r| 1> + < -1|r| 1>]
=21/2 < -1|r| 1>.
10. As another example of the use of the SC rules,
<| H|> = <|> = <*|*>
(note, again this is an exchange-type integral).
6. What is the Hamiltonian matrix element coupling | and
2-1/2 [ |*| - |*|]?
The first determinant differs from the 2 determinant by one spin-orbital, as does
the second (after it is pl
P 1/2 [|e1e2| +|e1e2|] = R (R) [R11R22 -R12R21 ]
1/2[|e1e2| +|e1e2|] .
The other (singlet) determinants can be symmetry analyzed in like fashion and result in the
P |e1e1| = R (R)cfw_R11R11|e1e1| +R12R12 |e2e2| +R11R12
and can be expressed as LCAO-MO's in terms of the individual pi orbitals as follows:
a1 =1/3 [ p1 + p2 + p3], e 1 = 1/2 [ p1 - p 3],
e2 = 1/6 [ 2 p2 -p1 -p3].
For the anion's lowest energy configuration, the orbital occupancy a12e2 must be
The only remaining entry, which thus has the highest MS and ML values, has MS = 0 and
ML = 0. Thus there is also a 1S level in the p2 configuration.
Thus, unlike the non-equiva
The 1D term symbol is handled in like fashion. Beginning with the ML = 2 state
|p1p1|, one applies L- to generate the ML = 1 state:
L- 1D(ML=2, M S =0) = [L-(1) + L-(2)] |p1p1|
= h(2(3)-2(1)1/2 1D(ML=1, M S =0)
= h(1(2)-1(0)1/2 [|p0p1| + |p1p0|],
It should be emphasized that the process of deleting or crossing off entries in various ML,
MS boxes involves only counting how many states there are; by no means do we identify
There is no need to form the corresponding states with negative ML or negative MS values
because they are simply "mirror images" of those listed above. For example, the state w
2. After identifying as many such states as possible by inspection, one uses L and S to
generate states that belong to the same term symbols as those already identified but which
have higher or lower ML and/or MS values.
3. If, after applying the above pr
and 1D functions because 1S, 3P, and 1D are eigenfunctions of the hermitian operator L2
having different eigenvalues. The state that is normalized and is a combination of p0p0|,
|p-1p1|, and |p1p-1| is given as follows:
= 3 -1/2 [ |p0p0| - |p-1p1| -
Recall that the symmetry labels e and o refer to the symmetries of the orbitals under
reflection through the one Cv plane that is preserved throughout the proposed disrotatory
closing. Low-energy configurations (assuming one is interested
In the study of Oscillatory motion, or periodic motion, experiments were performed by means of a falling body, a simple pendulum, an inertial balance, and by pendulum motion of walking. In the acceleration of a falling b