1. Electronic Structure Theory
Session 4
Jack Simons, Henry Eyring Scientist and Professor
Chemistry Department
University of Utah

2.   |he| > CJ, = J  <|> CJ,
Let’s get a bit more specific. How does one determine the orbitals J and then how does one determine the CI coefficients CJ?
The orbitals are usually determined by carrying out a HF calculation. This is not done (except in rare cases) by solving the HF second order partial differential equations in 3N dimensions on a spatial grid but by expanding the J in terms of so-called atomic orbital (AO) (because they usually are centered on atoms) basis functions using the LCAO-MO expansion:
J =   CJ,
This reduces the HF calculation to a matrix eigenvalue equation
  |he| > CJ, = J  <|> CJ,

3. + K=occ CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
The Fock-operator (F or he) matrix elements needed to carry out such a calculation are:
<| he| > = <| –2/2m 2 |> + a<| -Zae2/|ra |>
+ K=occ CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
– <(r) (r’) |(e2/|r-r’|) | (r) (r’)>]
and the overlap integrals:
<|>
Kinetic, Ve,e, Ve,n
The number of these one- and two electron integrals scales with the basis set size M as M2 and M4.
The computer effort needed to solve the MxM eigenvalue problem scales as M3.
The sum over K runs over all of the occupied spin-orbitals.
Recall this makes the occupied orbitals “feel” N-1 other electrons, but the virtual orbitals “feel” the N occupied spin-orbitals.

4. + K=occ CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
To form the elements of the MxM Fock matrix:
F =
<| he| > = <| –2/2m 2 |> + a<| -Zae2/|ra |>
+ K=occ CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
– <(r) (r’) |(e2/|r-r’|) | (r) (r’)>],
one needs to already know the LCAO-MO coefficients CK,for the occupied MOs.
A so-called self-consistent field (SCF) process is used to address this:

5. <| –2/2m 2 |> + a<| -Zae2/|ra |>
SCF:
One guesses (eigenfunctions of the Fock operator with all J and K terms ignored are often used, or coefficients from a calculation carried out at a “nearby geometry” are used) the CK,coefficients of the occupied spin-orbitals.
The Fock matrix is then formed using these CK,coefficients:
<| –2/2m 2 |> + a<| -Zae2/|ra |>
+ K=occ CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
– <(r) (r’) |(e2/|r-r’|) | (r) (r’)>]
The HF equations are solved to obtain “new” CK,coefficients:
  |he| > CJ, = J  <|> CJ,
These “new” CK,coefficients are used to form a “new” Fock matrix.
The HF equations are solved to obtain “newer” CK,coefficients.
This iterative solution is continued until the CK,coefficients used in one iteration are identical to those obtained in the next solution of the Fock matrix.
One has then achieved self-consistency.

6. It is crucial to understand that it is by “guessing” the initial values of the LCAO-MO coefficients that one specifies for which electronic state the HF-SCF spin-orbitals are to be obtained.
That is, one inputs the CK,coefficients of the N occupied spin-orbitals, then an MxM Fock matrix is formed and its M eigenvalues K and M eigenvectors CK,are obtained.
However, of the M spin-orbitals thus determined, only N are occupied.
One has to be very careful (often by visually examining the HF orbitals) that the spin-orbitals one wants occupied for the electronic state of interest are those included in the list of occupied spin-orbitals in each iteration of the SCF process. This is especially critical when studying excited states where the occupied spin-orbitals are probably not those having the lowest orbital energies K.

7. Suppose one were interested in studying an anionic state of formamide in which the excess electron occupies the OCN * orbital.
An SCF calculation on neutral formamide using an aug-cc-pVDZ basis set produces the orbitals shown below. The orbital energies for the bonding and non-bonding OCN  MOs (HOMO-2 and HOMO) are and eV, respectively. The HOMO-1 orbital is a lone pair orbital on the oxygen atom. The SCF orbital energy of the lowest unoccupied molecular orbital (LUMO) is 0.72 eV However, the LUMO is not even of * symmetry, nor is the LUMO+1 or the LUMO+2 orbital. The lowest unoccupied orbital of * character is the LUMO+3, and this orbital has an energy of eV.
So, to study formamide anion in its * state, one must “guess” the CK, coefficients of the LUMO+3 as an occupied MO.

8. Why UHF Wavefunctions are not eigenfunctions of S2
<| he| > = <| –2/2m 2 |> + a<| -Zae2/|ra |>
+ K CK, CK, [<(r) (r’) |(e2/|r-r’|) | (r) (r’)>
– <(r) (r’) |(e2/|r-r’|) | (r) (r’)>].
C: 1s22s22pz2py
The matrix elements of the Fock operator are different for an  and a  spin-orbital because the sum:
K CK, CK,
appearing in these matrix elements runs over all N of the occupied spin-orbitals.
When forming matrix elements for  type orbitals, there will be Coulomb integrals for K = 1s,1s,2s,2s, 2pz, and 2py and exchange contributions for K = 1s,2s, 2pz, and 2py.
On the other hand, when solving for spin-orbitals of  type, there will be Coulomb integrals for K = 1s,1s,2s,2s, 2pz, and 2py. But exchange contributions only for K =1s and 2s.

9. Here are the  (SOMO) and  (LUMO) orbitals of LiF–
 = – Ha
 = Ha

10. n,l,m (r,,) = Nn,l,m, Yl,m (,) rn-1 e-r
What are these atomic orbitals (AOs)?
Slater-type orbitals (STOs)
n,l,m (r,,) = Nn,l,m, Yl,m (,) rn-1 e-r
are characterized by quantum numbers n, l, and m and exponents (which characterize the radial 'size' )  and are usually located on one of the atomic nuclei.
Cartesian Gaussian-type orbitals (GTOs)
a,b,c (r,,) = N'a,b,c, xa yb zc exp(-r2),
are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents  which govern the radial 'size’.

11. <(r) (r’) |(e2/|r-r’|) | (r) (r’)>
Slater-type orbitals are similar to Hydrogenic orbitals in the regions close to the nuclei. Specifically, they have a non-zero slope near the nucleus on which they are located
(i.e., d/dr(exp(-r))r=0 = -,
so they can have proper electron-nucleus cusps.
In contrast, GTOs have zero slope near r=0 because
d/dr(exp(-r2))r=0 = 0.
This characteristic favors STOs over GTOs because we know that the correct solutions to the Schrödinger equation have such cusps at each nucleus of a molecule.
However, the multi-center integrals which arise in polyatomic-molecule calculations cannot efficiently be evaluated when STOs are employed. In contrast, such integrals can routinely be computed when GTOs are used. This advantage of GTOs has lead to the dominance of these functions in molecular quantum chemistry.
<(r) (r’) |(e2/|r-r’|) | (r) (r’)>

12. To overcome the cusp weakness of GTO functions, it is common to combine two, three, or more GTOs, with combination coefficients that are fixed and not treated as LCAO parameters, into new functions called contracted GTOs or CGTOs. However, does not really correctly produce a cusp because every Gaussian has a zero slope at r = 0 as shown below:

13. Most AO basis sets contain a mixture of different classes of functions.
Fundamental core and valence basis functions:
Polarization functions
Diffuse functions
Rydberg functions
Core and valence:
Minimal basis - the number of CGTOs equals the number of core and valence atomic orbitals in the atom.
Carbon - one tight s-type CGTO, one looser s-type CGTO and a set of three looser p-type CGTOs.

14. Minimal STO and GTO carbon 1s, 2, and 2p radial functions compared to the results obtained using a large AO basis.

15. Double-zeta (DZ) - twice as many CGTOs as there are core and valence atomic orbitals.
Carbon - two tight s, two looser s, and two sets of three looser p CGTOs.
Triple-zeta (TZ) - three times as many CGTOs as the number of core and valence atomic orbitals (extensions to quadruple-zeta and higher-zeta bases also exist).
The use of more basis functions is motivated by a desire to provide additional variational flexibility so the LCAO process can generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies.

16. Polarization functions:
one higher angular momentum than appears in the atom's valence orbital space.
d-functions for C, N, and O and p-functions for H with exponents ( or ) which cause their radial sizes to be similar to the sizes of the valence orbitals.
Note: the polarization p orbitals of H are similar in size to the valence 1s orbital and the polarization d orbitals of C are similar in size to the 2s and 2p orbitals, not like the valence d orbitals of C.
Polarization functions give angular flexibility to the LCAO process in forming molecular orbitals between from valence atomic orbitals.
Polarization functions also allow for angular correlations in describing the correlated motions of electrons.

17. s, p, d, f, and g angular functions showing how they span more and more of angle-space as L increases

18. The polarization functions also can be used to dynamically correlate electrons as in
 = C1 | .. ..| - C2 | ..' '..|
= C1/2 { | ..( - x') (  + x')..| - | ..( - x') (  + x')..| }
If  and ’ involve orbitals of different angular character (e.g., s and p, or d and p), one gains angular correlation.
If  and ’ involve orbitals of different radial character, one gains radial correlation.

19. An example of d polarization functions on C and O:

20. Core, valence and polarization functions do not provide enough radial flexibility to adequately describe very diffuse charge densities.
The diffuse basis functions tabulated, for example, on the PNNL web site ( are appropriate if the anion under study has its excess electron in a valence-type orbital (e.g., as in F-, OH-, carboxylates, etc.) but not for very weakly bound anions (e.g., having EAs of 0.1 eV or less).
For an electron in a Rydberg orbital, in an orbital centered on the positive site of a zwitterion species, or in a dipole-bound orbital, one must add to the bases containing valence, polarization, and conventional diffuse functions yet another set of functions that are extra diffuse. The exponents of these extra diffuse basis functions can be obtained by scaling the conventional diffuse functions’ smallest exponent (e.g, by 1/3).

21. An example of a species needing extra diffuse basis functions: Arginine anion