1. Electronic Structure Theory
Session 2
Jack Simons, Henry Eyring Scientist and Professor
Chemistry Department
University of Utah

2. 2-1[|RR|+ |LL|+ |RL|+ |LR|],
So, a single determinant may be qualitatively incorrect because it may contain ionic and diradical terms
|| =
2-1[|RR|+ |LL|+ |RL|+ |LR|],
or it may not be a proper spin eigenfunction |’|.
This remains a problem for any |(1)  (2)| case (
homopolar or heteropolar
In contrast, a one-electron bond |(1)| can dissociate correctly to
2-1/2 [L(1) + R(1)] as can |(1)| to 2-1/2 [L(1) - R(1)] if homopolar or |(1)|  |L(1)| and |(1)|  |R(1)|, if heteropolar
(L being more electronegative and R less)
The three-electron bonding and anti-bonding case is also OK:
|(1)  (2)  *(3)|  2-1/2[ |L(1)R(2)R(3)| -|R(1)L(2)L(3)|] for homopolar, or |L(1)L(2)R(3)| for heteropolar.
So, sometimes one can use a single Slater determinant as a reasonable starting point, but sometimes, one can not.

3. Recall that incorrect (ionic+diradical) bond dissociation can be “improved” by using a Slater determinant like |(1) ’ (2)|, but this can produce “jerks” in the energy profile and is not of pure spin.
Nevertheless, the single determinant forms the basis of RHF and UHF theory. So, let’s pursue this kind of function a bit further and see how these theories involve making specific choices for the VMF(r) potential.
First, we need to recall how to write matrix elements of one-and two- electron operators for determinant functions.

4. < p q | g | 'p 'q > - < p q | g | 'q 'p >
The Slater-Condon rules- memorize them, please
(i) If two determinants | > and | ' > are identical, then
< | F + G | > =
i < i | f | i > +i>j [< ij | g | ij > - < ij | g | ji >],
where the sums over i and j run over all spin-orbitals in | > and F and G are one- and two-electron additive operators;
(ii) If | > and | ' > differ by a single spin-orbital ( p  'p ),
< | F + G | ' > =
< p | f | 'p > +j [< pj | g | 'pj > - < pj | g | j'p >],
where the sum over j runs over all spin-orbitals in | > except p;
(iii) If | > and | ' > differ by two ( p  'p and q  'q),
< p q | g | 'p 'q > - < p q | g | 'q 'p >
(note that the F contribution vanishes in this case);
(iv) If | > and | ' > differ by three or more spin orbitals, then
< | F + G | ' > = 0;
(v) For the identity operator I, < | I | ' > = 0 if | > and | ' > differ by one or more spin-orbitals.

5. h J = J J = [Te + Ve,n + Vn,n] J
How does one find a good VMF? One way is to use the Single determinant trial function = |1 2 ...N|,
and write down < H > using the Slater-Condon rules (now, H denotes the electronic Hamiltonian; H0 will denote an approximation)
< H > = k=occ.< k|Te + Ve,n + Vn,n| k> + 1/2 k,l=occ.
[< k(1) l(2)|e2/r1,2| k(1) l(2)> - < k(1) l(2)|e2/r1,2| l(1) k(2)>]
and observe that Coulomb (Jk,l) and exchange (Kk,l) integrals among occupied
spin-orbitals arise.
If one minimizes this energy with respect to the J’s, with the constraint that
< J| K> = J,K,
one obtains the HF equations:
h J = J J = [Te + Ve,n + Vn,n] J
+ k [< k(1)|e2/r1,2| k(1)> J(2) - < k(1)|e2/r1,2| J(1)> k(2)
that contain the Jk and Kk potentials.

6. J1,2=  |1(r)|2 e2/|r-r’||2(r’)|2 dr dr’ =  J1(r’) |2(r’)|2 dr’
A physical picture of Coulomb and exchange interactions:
J1,2=  |1(r)|2 e2/|r-r’||2(r’)|2 dr dr’ =  J1(r’) |2(r’)|2 dr’
=  |1(r)|2 J2(r) dr
K1,2=  1(r) 2(r’) e2/|r-r’|2(r) 1(r’)dr dr’
[Te + Ve,n + Vn,n] J
Involves the kinetic, electron-nuclei, and nuclear-nuclear energies.

7. <H  > = <H0  >, if H0 is defined as
So, one can define VMF in terms of the J and K interactions. This is the Hartree-Fock definition of VMF.
It has the characteristic that
<H  > = <H0  >,
if H0 is defined as
H0 = Te + Ve,n + Vn,n + (J-K).
In this case,
H-H0 = Ve,e – (J-K).
There is no first-order perturbation correction to the energy because
<H  > = <H0  >.
This choice of H0 forms the basis of Møller-Plesset perturbation theory (MPn).
Notice that it is by making a mean-field model that our (chemists’) concepts of orbitals J and of electronic configurations (e.g., 1s 1s  2s  2s  2p1 ) arise.

8. ||k=occ.< k|Te + Ve,n + Vn,n| k> + 1/2 k,l=occ.
A good thing about the HF VMF is that the orbital energies K give approximate ionization potentials and electron affinities (Koopmans’ theorem). This can be derived by writing down the energies of two Slater determinants
0 = |1 2 ...N|
and
- = |1 2 ...N N+1| ,
using
||k=occ.< k|Te + Ve,n + Vn,n| k> + 1/2 k,l=occ.
[< k(1) l(2)|e2/r1,2| k(1) l(2)> - < k(1) l(2)|e2/r1,2| l(1) k(2)>]
and subtracting the two energies to obtain the energy difference.
E = < N+1|Te + Ve,n + Vn,n| N+1>
+ l [< N+1 (1) l(2)|e2/r1,2| N+1 (1) l(2)>
- < N+1 (1) l(2)|e2/r1,2| l(1) N+1 (2)>] = N+1

9. < o|Te + Ve,n + Vn,n| o>
It is important to notice (and remember!) that the occupied orbitals’
< o|Te + Ve,n + Vn,n| o>
+ l [< o (1) l(2)|e2/r1,2| o (1) l(2)>
– < o (1) l(2)|e2/r1,2| l(1) o (2)>] = o
feel Coulomb and exchange interactions with the N-1 “other” occupied orbitals because the l=o terms cancel.
However, the unoccupied (virtual) orbitals “feel” all N occupied orbitals:
< u|Te + Ve,n + Vn,n| u> + l [< u (1) l(2)|e2/r1,2| u (1) l(2)>
– < u (1) l(2)|e2/r1,2| l(1) u (2)>] = u

10. So, within the limits of Koopmans’ theorem,
-u gives an approximation to the energy gained by adding an electron to
u the EA
-o gives an approximation to the energy gained by removing an electron from
o the IP
u -o should not be used as an approximation to the energy gained by promoting an electron from o to u because promotion leaves behind a hole in o and -u is the energy gained when one puts an electron into u but not with a hole in o.

11. ||k=occ.< k|Te + Ve,n + Vn,n| k> + 1/2 k,l
Another thing to be aware of is that the sum of the occupied orbital energies is not equal to the HF energy:
||k=occ.< k|Te + Ve,n + Vn,n| k> + 1/2 k,l
[< k(1) l(2)|e2/r1,2| k(1) l(2)> - < k(1) l(2)|e2/r1,2| l(1) k(2)>]
k=occ. k = k < k|Te + Ve,n + Vn,n| k> + 1 l,k
The sum of orbital energies double counts the J-K interactions.
So, one has to be careful to understand what orbital energies do and do not mean.

12. It is also important to realize that the HF orbitals tend to be more delocalized than one is usually shown in textbooks.
Here are some P=O * and C-O * orbitals for a O=P(OH)(O-R)(O-R’) compound consisting of a phosphate group bonded to two sugar moities.

13. Another property of HF orbitals is that, for them, the Brillouin theorem holds. This says the the matrix element of H connecting the HF Slater determinant to a Slater determinant in which any one occupied spin-orbital a is replaced by a virtual spin-orbital m vanishes.
< |1 2 a...N| H |1 2 m...N| > = <a| Te + Ve,n + Vn,n|m> + l
[< a(1) l(2)|e2/r1,2| m(1) l(2)> - < a (1) l(2)|e2/r1,2| l(1) m(2)>]
= <a|hHF | m> = 0
The importance of this is that, if one uses the HF determinant as a zeroth-order wave function, 0, then will not contain any singly excited determinants; the lowest-level excitations in  will be doubly excited determinants.

14. Configuration State Functions
Some single-configuration functions can not be single determinants.
There are cases where more than one determinant must be used. Although the determinant |1s 1s 2s 2s 2pz 2py| is an acceptable approximation to the carbon 3P state if the 1s and 2s spin-orbitals are restricted to be equal for  and  spins, the 1S state arising in this same 1s22s22p2 configuration can not be represented as a single determinant.
The 1S state requires the following three-determinant function:
 = 3-1/2 [1s 1s 2s 2s 2pz 2pz|
- 1s 1s 2s 2s 2px 2px| - 1s 1s 2s 2s 2py 2py| ].
If a state cannot be represented by a single determinant, one should not use theoretical methods that are predicated on a dominant single determinant in the expansion of the full wave function!