1. Orbitals, Basis Sets and Other Topics
Ode to Slater

2. Improving the zero-order (aka one-electron) approximation
By ignoring the electron-electron repulsion terms, while leaving the attractive terms unaltered, we are doomed to obtain a result that is far too stable.
We cannot insert the repulsion terms without destroying the one-electron approximation.
To improve the result, we must reduce the magnitude of the attractive interactions to a level that is more reasonable.

3. Shielding
As electrons are added farther from the nucleus, inner electrons shield the outer ones from some of the nuclear charge.
Electrons with larger values of n have small maxima in their probability distributions inside the nodal surfaces and these affect the shielding of nuclear charge at larger radial distances.
In general, for a given value of n, s electrons shield more than p electrons etc.

4. Slater’s Rules
J. C. Slater was responsible for a set of rules, Slater’s Rules, that permit calculation of the effective nuclear charge, Zeff = Z - s, that makes allowance for shielding.
The shielding effects of orbitals are grouped as follows: (1s), (2s, 2p), (3s,3p), (3d), (4s,4p), (4d), (4f), (5s,5p), etc…
Electrons in orbitals at the right in this list do not shield those to the left.

5. For ns and np electrons
Electrons in the same grouping contribute 0.35 to s except for 1s which contributes 0.30.
Electrons in the next lower shell (n-1) contribute 0.85 to s.
Electrons below this (n-2, etc.) contribute 1.00 to s.

6. For nd and nf electrons
Electrons in the same nd OR nf contribute 0.35 to s.
Electrons below this (to the left of) contribute 1.00 to s.
To rationalize this, consider that, e.g., for a 4f electron, the valence shell is 6 (6s orbitals fill before 4f according to the Aufbau principle) so a 4d electron is two n values below the n-value of the valence shell, thus we apply 1.0 to the screening constant.

7. In-class problem
Using Slater’s rules, determine Zeff for a 2p electron in N, O, and F.

8. Slater-type orbitals (STOs)
Modify the hydrogenic eigenfunctions to account for effective nuclear charge. Radial part is also modified to retain only highest power in r in polynomial:

9. More accurate Zeff
Generally obtained by requiring that they produce the most accurate possible result in a variational calculation of the ground-state energy, using STOs.
Seminal calculations by Clementi and Raimondi for 2  Z  86.

10. Example
Using the given table, obtain the form of a normalized Slater 3s orbital for the Na atom.

11. Comparison to hydrogenic orbitals
Similar at r > 4 bohr.
Nodal structure lost.

12. Radial probability density
Again, no nodal structure.
STO does not extend to larger values of r.

13. Computational shortcoming
From a computational point of view the STOs have the severe shortcoming that most of the required integrals needed in the course of the SCF procedure must be calculated numerically which drastically decreases the speed of a computation.

14. Gaussian-type orbitals (GTOs)
Same basic form as STOs, except the exponential term is dependent on r2
The main difference to the STOs is that the variable r in the exponential function is squared. This leads to a vanishing slope at the center of the GTOs instead of a cusp with finite slope.

15. Example- H atom For n=1, the normalized Gaussian orbital is
Evaluation of the energy expression leads to

16. Minimize E
Solving the minimization problem E/=0 yields =0.2827, and E = 0.424 hartrees.
True E = 0.500 hartrees; %error = 15.2%.
Using a linear combination of 2 GTOs and varying  gives a slight improvement, to E = 0.436 hartrees.

17. The gaussian function is rounded off near the nucleus (rather than peaked, as in the true wavefunction), and does not behave properly at large distances from the nucleus.

18. Improving Gaussians
Use linear combination of 2 GTOs with two different variational parameters.
Only slight improvement in energy, at considerable computational cost.
Use a linear combination of simple gaussian functions to form approximate representation of true 1s function. By use of enough terms, exact wavefunction can be approximated to any desired accuracy.

19. Computational advantage
The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them.

20. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4--5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

21. Basis Functions Hydrogenlike orbitals Advantage: orthogonal
Requires infinite number to be complete
Difficult calculations
Disadvantages:

22. Basis Functions Slater-type orbitals (STO) Advantage:
Finite set is complete
Advantage:
Difficult calculations
Disadvantages:
Nonorthogonal

23. Basis Functions Gaussian-type orbitals (GTO) Advantages:
Finite set is complete
Advantages:
Easy/Fast calculations
Poor at representing electron close to nucleus and far away
Disadvantages:
Nonorthogonal
Calculation speed means can use many more GTO than STO functions and still be faster

24. Electron Exchange Symmetry
When studying the He atom, we saw that because the electrons must be antisymmetric with respect to interchange, of the four possible spin functions
    only the 1st two are valid, with linear combinations required for the other two.
We now develop a general method to write wavefunctions with required antisymmetry property.

25. Ground-state He wavefunctions
Let
with a similar definition for
A valid wavefunction for He would be
The (1,2) notation signifies that the wavefunction depends on the coordinates of both Electron 1 and Electron 2.

26. Determinants
When two rows or two columns of a determinant are exchanged, magnitude remains unaltered, but sign changes.
Example:
det

27. Slater determinant
Put two different one-electron functions in different rows; assign different electrons to different columns.
Electron 1 only in column 1, Electron 2 only in column 2.

28. Column exchange = electron exchange
Expand Slater determinant:
Result: symmetric spatial part, antisymmetric spin function, antisymmetric wavefunction.

29. Notes Only a two-electron wavefunction has the latter property.
N-electron atom requires N spin orbitals, N  N Slater determinant.
Normalization constant for an N  N Slater determinant will be [1/N!]1/2

30. Example
Let 1s be a Slater-type orbital with the effective charge for He given in the handout table.
Write down the form of this orbital and normalize it.
Obtain the normalization constant for the Slater determinant.

31. Pauli Exclusion Principle
Attempt to assign same spin orbital to each electron in He.
When any two rows or columns of a determinant are identical, the determinant is always zero. This produces  = 0.