Restricted Hartree–Fock for Beryllium

Part 1 Function Definition

Define STO function

$STO$$\; function \; has \; format \; of: \; Nr^{n-1}e^{-r\zeta}$

Overlap Integrate

$S = \int_{0}^\infty f_1^* f_2 \; r^2dr$

Hamiltonian core

H core = kinetics energy + electron and nuclear potential energy

$H = \int_{0}^\infty f_1 \hat{H} f_2 \; r^2dr$

$H = \int_{0}^\infty f_1 ((-\dfrac{1}{2}) \nabla^2 - \dfrac{Z_{\alpha}}{r})f_2 \; r^2 dr$

$Laplace \; operator:$ $\nabla^2$ $\therefore H = \int_{0}^\infty f_1 ((-\dfrac{1}{2}) \dfrac{1}{r} \dfrac{\partial}{\partial r} \dfrac{\partial}{\partial r} r f_2 - \dfrac{Z_{\alpha}}{r}r^2 f_2 )dr$

reminder: change the Z of nucleus if you want to run for other atom

Two elecron repulsion integral

$(rs|tu) = \int \int \dfrac{f_r^*(1) f_s(1) f_t^*(2) f_u(2)}{r_{12}} \; dv_1 dv_2$

For 1s 2s orbital

$(rs|tu) = \int_{0}^\infty \int_{0}^\infty \dfrac{f_r^*(1) f_s(1) f_t^*(2) f_u(2)}{r_{12}} \; r_1^2dr_1\; r_2^2dr_2$

$(rs|tu) = \int_{0}^\infty f_r^*(1) f_s(1) \; r_1^2dr_1\int_{0}^\infty \frac{ f_t^*(2) f_u(2)}{r_{12}}\; r_2^2dr_2$

From problem 9.14 in quantum_chemistry by levine

$(rs|tu) = \int_{0}^\infty f_r^*(1) f_s(1) \; r_1^2dr_1\int_{0}^\infty \frac{ f_t^*(2) f_u(2)}{r_{>}}\; r_2^2dr_2$

$(rs|tu) = \int_{0}^\infty f_r^*(1) f_s(1) \; r_1^2dr_1(\int_{0}^{r_1} \frac{ f_t^*(2) f_u(2)}{r_{1}}\; r_2^2dr_2 + \int_{r_1}^\infty \frac{ f_t^*(2) f_u(2)}{r_{2}}\; r_2^2dr_2)$

$Let \; B= \int_{0}^{r_1} \frac{ f_t^*(2) f_u(2)}{r_{1}}\; r_2^2dr_2 + \int_{r_1}^\infty \frac{ f_t^*(2) f_u(2)}{r_{2}}\; r_2^2dr_2$

Density matrix

$P_{tu} =2 \sum_{j=1}^{n/2}c_{tj}^* c_{uj}$

Reminder: P need to be changed if the atom have unpaired electron

Fock matrix

$F_{rs} = H_{rs}^{core} + \sum_{t=1}^{b} \sum_{t=1}^{b}P_{tu}[(rs|tu)- \frac{1}2(ru|ts)]$

$G = \sum_{t=1}^{b} \sum_{t=1}^{b}P_{tu}[(rs|tu)- \frac{1}2(ru|ts)]$

$F_{rs} = H_{rs}^{core} + G$

$In\;G \;one\;is\;coulombic\;repulsion,\;another\;is\;exchange\;energy$

Solve Hartree-Fork equation

$det(F_{rs}-\epsilon_i S_{rs} = 0)$

$The\;energy\;returned\;is\;the\;orbital\;energy\;for\;1\;electron$

Return atom energy

$E_{HF} = \sum_{i=1}^{2/n}\epsilon +\frac{1}2 \sum_{r=1}^{b} \sum_{s=1}^{b}P_{rs}H_{rs}+V_{NN}$

Part 2 Hartree Fork Iteration

• Initialization
1. initializing Co (coefficients) without considering electron repulsion
2. Solve Hartree-Fork equation with H_matrix and S_matrix to get initial Co
• Iteration
1. Using Co, we can get P_matrix (electron density)
2. Using P_matrix, H_matrix, G_matrix => F_matrix
3. Solve Hartree-Fork equation with F_matrix and S_matrix to get improved orbital energy and Co, which also means improved orbital functions.
4. Using improved Co, return to step 1