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.
Chapter 12
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).
When
< |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:
< 2p02p0
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
and
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
*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
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
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
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);
two-electron
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
Chapter 11
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,
and
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
following:
P |e1e1| = R (R)cfw_R11R11|e1e1| +R12R12 |e2e2| +R11R12
[|e1e2|-|e1e2|],
P |e
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],
and
e2 = 1/6 [ 2 p2 -p1 -p3].
For the anion's lowest energy configuration, the orbital occupancy a12e2 must be
considered,
orbital generally yields a linear combination of the degenerate orbitals rather than a multiple
of the original orbital (i.e., R i = i(R) i is no longer valid). For example, when a pair of
degenerate orbitals (denoted e1 and e2 ) are involved, one has
R e
After the unique entries of the box have been identified, one uses S operations to
find the other functions. For example, the wavefunctions of the 3A2 and 1A2 states of the
n* 1a122a123a121b224a121b125a122b212b11 configuration of formaldehyde are formed b
orbitals is used to obtain the symmetry of the product wavefunction: A2 =b1 x b2). The
=> * excited configuration 1b112b11 gives 1A1 and 3A1 states because b1 x b1 = A1.
The only angular momentum coupling that occurs in non-linear molecules involves
the
arrive at the total ML value. This angular momentum coupling approach gives the same set
of symmetry labels ( 3, 1, 3, 1, 3, and 1 ) as are obtained by considering all of the
determinants of the composite system as treated above.
IV. Non-Linear Molecule T
For a 1' 1 configuration in which two non-equivalent orbitals (i.e., orbitals
which are of symmetry but which are not both members of the same degenerate set; an
example would be the and * orbitals in the B2 molecule) are occupied, the above
analysis must
As with the atomic systems, additional examples are provided later in this chapter and in
Appendix G.
D. Inversion Symmetry and v Reflection Symmetry
For homonuclear molecules (e.g., O2, N 2, etc.) the inversion operator i (where
inversion of all electron
Recognizing that v *1 = *-1 and v *-1= *1, then gives
v |S=1, M S =1> = |*1*-1|
= 2-1/2 [ *-1(r1 ) 1 *1(r2 ) 2 - *-1(r2 ) 2 *1(r1 ) 1 ]
= (-1) 2-1/2 [ *1(r1 ) 1 *-1(r2 ) 2 - *1(r2 ) 2 *-1(r1 ) 1 ],
so this wavefunction is odd under v which is written as 3
whenever one is faced with equivalent angular momenta in a linear-molecule case, one must
use the box method to determine the allowed term symbols. If one has a mixture of
equivalent and non-equivalent angular momenta, it is possible to treat the equivale