Solutions of Schrödinger’s Equation

STOs vs GTOs

The wavefunction consists of two parts, a radial part(\(r\)) and a spherical harmonic part(\(\theta,\varphi\))

\[ \phi_{n,l,m}(r,\theta,\varphi)=\frac{\xi^3}{\sqrt{\pi}}Y^l_m(\theta,\varphi)\cdot r^n\cdot e^{-\zeta r} \]

Spherical harmonic determines the shape of the orbital (s,p,d,f) and the radial component determines how far the orbital extends form the nuclei.

STOs

Are an exact analytical solution to the hydrogen atom wavefunction and have a non-zero derivative at the nucleus (this is an issue). Since the viriol theorem tells us that the electrons don’t collapse on to the nucleus, as the wavefunction reaches \(r=0\) we need to change the energy to \(=-z\psi\), where \(z=\) nuclear charge. in STOs, the \(\zeta\) controls the width of the wavefunction.

GTOs

GTOs are a finite function an has a zero derivative a the nucleus. GTOs can’t accurately represent STOs, so we need 3 to represent one properly. This till has a much wider extent which can become an issue for UV-Vis calculation.

\[ \phi_{n,l,m}(r,\theta,\varphi)=\frac{2\xi^3}{^4\sqrt{\pi^3}}Y^l_m(\theta,\varphi)\cdot r^n\cdot e^{-\zeta r^2} \]

Integrating the Orbitals

When integrating the orbitals, \(r\) has three coordinates (cartesian), as such:

\[ r=\sqrt{x^2+y^2+z^2} \]

For STOs

\[ e^{-\zeta\cdot r}=e^{-\zeta\cdot\sqrt{x^2+y^2+z^2}} \]

This results in the exponential of a square root which is computationally expensive.

For GTOs

\[ e^{-\zeta\cdot r^2}=e^{-\zeta\cdot(x^2+y^2+z^2)}=e^{-\zeta\cdot x^2}\cdot e^{-\zeta\cdot y^2}\cdot e^{-\zeta\cdot z^2} \]

This is much easier to calculate, as \(\int_{-\infin}^{+\infin}e^{-x^2}dx=\sqrt{\pi}\) is a definite integral.

Basis Sets (\(\chi_j\))

are a selection of one electron functions for the Schrodinger equation. These are the eigenvectors of the Hamiltonian , which means that the most simple solution is the atomic orbitals (\(\psi_i\)) that comprise the wavefunction.

LCAO states that the sum of the basis functions gives us the atomic orbital and that the sum of the atomic orbitals gives the molecular orbitals, so the sum of the basis functions gives the wavefunction.

\[ \psi_i=\sum\limits_{j=1}^{N}d_{ij}\chi_j \]

Minimal Basis Sets

Are simply calculated as one function per orbital

All Electron Basis Sets

Treat all the electrons in the system with the same number of basis functions

Split Valence Basis Sets

Treat the core electrons differently to the valence electrons and use the weighted sum of the basis functions to build the molecular orbital.

E.g. for double zeta:

\[ \psi_{2s}(r )=C_1\cdot \psi_{2s}(r,\zeta_1)+C_2\cdot\psi_{2s}(r,\zeta_2) \]

Polarisation Functions

Add in extra functions to account for a higher angular momentum than is currently occupied. This has the function of re-shaping the original AOs. this is primarily needed for:

• Bond angles
• Dipole moments
• Rotation barriers within the molecules

Diffuse Functions

These add functions of lower angular momentum to make the resulting AO bigger. these are needed for:

• Polar bonds
• Anions and EWGs
• Weak intermolecular interactions (especially dispersion)
• Raman intensities and polarisabilities
• Excitation energies and Rydberg states

Dunning’s Correlation Consistent Basis Sets

Are calculated using highly correlated levels of theory, on free atoms and are called “consistent” because they consistently add one basis function for each additional with each additional zeta.

E.g.:

DZ adds SZ+1s+1p+1d TZ adds DZ+1s+1p+1d+1f

They are very good at recovering correlation energy, double zeta can recover 65%, triple zeta can recover 85% and quadruple zeta can recover 93%

Infinite Basis Functions

Theoretically, an infinite amount of basis functions will perfectly represent the true wavefunction of the system, though we practically need to truncate the basis set somewhere and will thus need to make choices of what to prioritise.

The Hartree Fock Method

The HF method works by separating the many-electron system into lots of individual, single-electron solutions, with an individual Fock operator (\(\hat{f}(i)\)) which is made up of the one electron Hamiltonian (\(\hat{h}(i)=\hat T_e(\vec r)+V_{ne}(\vec r,\vec R)\)) and the mean field (\(v^{HF}\)), which is the sum of the interaction energy of all the electrons in the system.

\[ \hat{f}(i)=\hat{h}(i)+v^{HF}(i) \]

The mean field is approximated through the Coulomb (\(j_{ij}\)) and exchange (\(k_{ij}\)) operators. The Coulomb operator calculates the electron repulsion between two electrons.

\[ j_{ij}=\left\langle\varphi_i(1)\varphi_j(2)\left|\frac{1}{r_{12}}\right|\varphi_i(1)\varphi_j(2)\right\rangle \]

The exchange operator accounts for electrons being indistinguishable and being able to swap positions with each other. This contributes a very small amount to the correlation energy and so HF is still considered to be correlation free,

\[ k_{ij}=\left\langle\varphi_i(1)\varphi_j(2)\left|\frac{1}{r_{12}}\right|\varphi_i(2)\varphi_j(1)\right\rangle \]

HF Energy

HF energy is the sum of the orbital energy (\(\epsilon\)) with adjustments from the exchange and coulombic operators.

\[ E(HF)=\sum\limits_{i=1}^{N}\epsilon+\frac{1}{2}J-K \]

The Fock equation returns three components

• The HF energy

• The energies of all the orbitals (occupied and unoccupied)

• All the MOs from the basis sets

MOs are not real

It’s worth being aware that MOs are not a real entity. They are the product basis sets of the QM calculation. The nature of having discrete electrons in discrete orbitals breaks the fundamental principles of quantum mechanics that state that electrons don’t really exist as particles. While we can get an idea of the electron density an we can use MOs to understand some properties of reactivity, they are fundamentally not something real.

RHF, ROHF and UHF

RHF is appropriate for closed shell systems, where all the electrons are paired. for UHF, the SCF is calculated once for \(\alpha\) (spin up) orbitals and a second time for \(\beta\) (spin down) orbitals. Since there will be a different amount of electrons in each, the resulting energies will be different between the \(\alpha\) and \(\beta\) orbitals.

Spin Contamination

This error is known as “spin contamination” and can be quantified to detemrine how bad it is. The QM package will spit out a value of the spin contamination (\(\langle s^2\rangle\)) that we can compare to our theoretical limit.

\[ \text{theoretical limit}=S(S+1) \]

Where S = the total spin of the system.

If \(\langle s^2\rangle\approx\) theoretical limit, the calculation is fine

If \(\langle s^2\rangle>>\) theoretical limit, you need to use ROHF

ROHF

ROHF calculates the paired electrons together and the unpaired electrons separately. While this will fix the spin contamination issue, it will render the orbital energy useless.

Method	RHF	ROHF	UHF
For systems with Multiplicity:	1	Any	Any
Do \(\alpha\) and \(\beta\) orbitals have the same spacial component	Yes	Yes	For multiplicity 1 yes, otherwise no
Relative energy for multiplicity 1	Identical	Identical	Identical
Relative energy for multiplicity >1	N/A	Higher than E(UHF)	Lower than E(ROHF)
Advantages (multiplicity > 1)	N/A	Simple orbital analysis Spin of the system is correct	Energy is lower than E(ROHF)
Disadvantages (multiplicity > 1)	N/A	Energy is higher than E(RHF)	Spin of the system is not correct

For Radicals

When doing calculations with radicals (including reactions) RHF may not work. In a water dissociation:

\[ \ce{H2O->HO. + H.} \]

However RHF will pair all the electrons, resulting in:

\[ \ce{H2O-> HO- + H+} \]

UHF will probably also not work for energy calculations, as there will be a large amount of spin contamination for anything beyond \(\ce{H2}\) or \(\ce{H2+}\).

Note

This also means that MP2 cannot e used for radicals, as it fundamentally uses RHF.