20_01fig_PChem.jpg Hydrogen Atom M m r Potential Energy + Kinetic Energy R C

20_01fig_PChem.jpg Hydrogen Atom RadialAngular Coulombic

20_01fig_PChem.jpg Hydrogen Atom will be an eigenfunction of Separable

20_01fig_PChem.jpg Hydrogen Atom Recall Bohr Radius

20_01fig_PChem.jpg Hydrogen Atom Assume Let’s try It is a ground state as it has no nodes

20_01fig_PChem.jpg Hydrogen Atom The ground state as it has no nodes n=1, and since l =0 and m = 0, the wavefunction will have no angular dependence

20_01fig_PChem.jpg Hydrogen Atom In general: Laguerre Polynomials 1S- 0 nodes 2S- 1 node 3S-2 nodes

Energies of the Hydrogen Atom In general: Hartrees kJ/mol

Wave functions of the Hydrogen Atom In general: Z=1, n = 1, l = 0, and m = 0:

Z=1, n = 2, l = 0, and m = 0: Wave functions of the Hydrogen Atom

Hydrogen Atom Z=1, n = 2, l = 1 m = 0:m = +1/-1: + _

20_06fig_PChem.jpg For radial distribution functions we integrate over all angles only Prob. density as a function of r. Radial Distribution Functions

20_09fig_PChem.jpg Radial Distribution Functions

20_08fig_PChem.jpg X Y Z Probability Distributions

20_12fig_PChem.jpg Atomic Units Set: Hartrees a.u. Much simpler forms.

Atoms Potential Energy Kinetic Energy C meme meme =r 12 M

Helium Atom C meme meme =r 12 M Cannot be separated!!! Hydrogen like 1 e’ Hamiltonian i.e. r 12 cannot be expressed as a function of just r 1 or just r 2 What kind of approximations can be made?

Ground State Energy of Helium Atom EoEo E1E1 E2E2 I 1 = ev EoEo E1E1 E2E2 I 2 = ev Ionization Energy of He E Free E o = ev = ev = Hartrees Perturbation Theory

Ground State Energy of Helium Atom H Not even close. Off by 1.1 H, or 3000 kJ/mol Therefore e’-e’ correlation, V ee, is very significant

Ground State Energy of Helium Atom

Closer but still far off!!! Perturbation is too large for PT to be accurate, much higher corrections would be required

Variational Method The wavefunction can be optimized to the system to make it more suitable Consider a trail wavefunction and Is the true wavefunction, where: Then The exact energy is a lower bound is a complete set Assume the trial function can be expressed in terms of the exact functions We need to show that

Variational Method Since Variational Energy E0E0 E var (  ) min

Variational Method For He Atom Let’s optimize the value of Z, since the presence of a second electrons shields the nucleus, effectively lowering its charge.

Variational Method For He Atom

Similarly Recall from PT

Variational Method For He Atom Much closer to H (  E= H =  kJ/mol error)

Variational Method For He Atom Optimized wavefunction

Variational Method For He Atom Optimized wavefunction Other Trail Functions (  E= H =  kJ/mol error) Optimizes both nuclear charges simultaneously

Variational Method For He Atom Other Trail Functions (  E= H =  kJ/mol error) Z’, b are optimized. Accounts for dependence on r 12. In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required. The more accurate the energy the more parameters that must be optimized the more demanding the calculation.

Variational Method For He Atom In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required. The more accurate the energy the more parameters that must be optimized the more demanding the calculation H H H Experimental ev H

The H 2 + Molecule One electron problem Two nuclei Define electron position, i..e. internal coordinates, w.r.t. nuclear positions.

The H 2 + Molecule Since Z A =1 and Z B =1

The H 2 + Molecule Nuclear Electronic The nuclear positions determine the electronic wavefunction Assume electronic motion is much faster than nuclear motion, implies that the nuclear positions are essentially static The electronic part is determined by the nuclear positions Separable?? W- Total Energy

The H 2 + Molecule Potential energy surface. Of primary interest Nuclear Electronic

Linear Variational WFctns. Suppose the trial wavefunction can be expressed in terms of an expansion of an appropriate set of functions, not necessarily othonormal

Linear Variational WFctns. For each c i Find the optimum coefficients, that minimize E var. 1 1

Linear Variational WFctns. Need to diagonalize matrix, to find eigenvalues and eigen vectors:

Linear Combination of Atomic Orbitals. Lets use the 1s Hydrogen like orbitals to be a basis for a trial function and apply variational theory to find the best approximate wavefunction Whereare Hydrogen like wavefunction with n=1, l=0, centred in nucleus a and b, resp.

Linear Combination of Atomic Orbitals.

Prediction of the Bond

Bonding and Antibonding Orbitals of H 2 +

23_09fig_PChem.jpg Density Difference Between MO’s and 1s O’s

23_11fig_PChem.jpg Electron Densities of Sigma and Pi M.O’.s Bonding Antibonding g=gerade (same) u=ungerade (opposite)

-13.6 e.v e.v e.v. Electron population on F is larger, ie. bond in polarized to F, ie. shows the F is more electronegative. Other Types of M.O.’s

23_13fig_PChem.jpg MO’s for the Diatomics

23_02tbl_PChem.jpg Energy Level Diagram For the Diatomics Electron Configuration for H 2 &He 2

23_17fig_PChem.jpg Electron Configuration of N 2

23_16fig_PChem.jpg Electron Configuration of F 2

23_18fig_PChem.jpg Electron Configurations of the Diatomics

23_20fig_PChem.jpg Bonding in HF