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Computational Chemistry

Field: Quantum Chemistry

A branch of chemistry that uses computer simulation to assist in solving chemical problems.

Sequence of Expressions

Definition

Born's Rule

For a normalized wavefunction Ψ(r)\Psi(\mathbf{r}), the probability density ρ(r)\rho(\mathbf{r}) of finding the particle at position r\mathbf{r} is given by the square of the magnitude of the wavefunction:\nρ(r)=∣Ψ(r)∣2\rho(\mathbf{r}) = |\Psi(\mathbf{r})|^2
Definition

Wavefunction

Let H^\hat{H} be the Hamiltonian operator for a system of NN particles, and let Ψ(r1,…,rN)\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N) be the wavefunction. The wavefunction must satisfy the time-independent Schrödinger equation:\nH^Ψ=EΨ\hat{H}\Psi = E\Psi\nwhere H^=∑i=1N(−ℏ22mi∇i2+V(ri))\hat{H} = \sum_{i=1}^{N} \left(-\frac{\hbar^2}{2m_i}\nabla_i^2 + V(\mathbf{r}_i)\right) is the Hamiltonian, and EE is the total energy eigenvalue.
Let Ψ(r,t)\Psi(\mathbf{r}, t) be the time-dependent wavefunction of a quantum system, and H^\hat{H} be the Hamiltonian operator. The evolution of Ψ\Psi is governed by the equation:\nH^Ψ(r,t)=iℏ∂∂tΨ(r,t)\hat{H}\Psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}, t)
Define the Hamiltonian operator H^\hat{H} for a system of particles with mass mkm_k and position rk\mathbf{r}_k as the sum of the kinetic energy operator T^\hat{T} and the potential energy operator V^\hat{V}: \nH^=T^+V^\hat{H} = \hat{T} + \hat{V} \nwhere T^=∑kpk22mk\hat{T} = \sum_{k} \frac{\mathbf{p}_k^2}{2m_k} and V^=∑k<le24πϵ0rkl+Vext(r1,…,rN)\hat{V} = \sum_{k<l} \frac{e^2}{4\pi\epsilon_0\mathbf{r}_{kl}} + V_{ext}(\mathbf{r}_1, \dots, \mathbf{r}_N). Here, pk\mathbf{p}_k is the momentum operator, pk=−iℏ∇k\mathbf{p}_k = -i\hbar\nabla_k.
Consider a particle of mass mm encountering a potential barrier V(x)V(x) defined over x∈[a,b]x \in [a, b], such that E<V(x)E < V(x) in this region. The transmission probability TT through the barrier, derived from the WKB approximation, is given by:\nT≈exp⁡(−2ℏ∫ab2m(V(x)−E) dx)T \approx \exp\left(-\frac{2}{\hbar} \int_{a}^{b} \sqrt{2m(V(x) - E)} \, dx\right)\nwhere EE is the particle's total energy.
For a single vibrational mode of a diatomic molecule with reduced mass μ\mu and force constant kk, the system is modeled by the harmonic oscillator Hamiltonian H^HO\hat{H}_{HO}: \nH^HO=p^22μ+12kx^2\hat{H}_{HO} = \frac{\hat{p}^2}{2\mu} + \frac{1}{2}k\hat{x}^2\nThe minimum possible energy, or Zero-Point Energy (ZPE), is the ground state eigenvalue (v=0v=0) of this Hamiltonian:\nEZPE=12ℏω=12ℏkμE_{ZPE} = \frac{1}{2} \hbar \omega = \frac{1}{2} \hbar \sqrt{\frac{k}{\mu}}
The total wavefunction Ψ\Psi is approximated by a single Slater determinant formed from orthonormal single-particle orbitals ϕi\phi_i: Ψ=det⁡(ϕ1,…,ϕN)\Psi = \det(\phi_1, \dots, \phi_N). The resulting single-particle equations are solved variationally, leading to the Roothaan-Hall equations in the basis set representation: \n∑j=1N(Hij−SijE)cj=0\sum_{j=1}^{N} (\mathbf{H}_{ij} - \mathbf{S}_{ij} \mathbf{E}) c_{j} = 0 \nwhere H\mathbf{H} and S\mathbf{S} are the Fock and overlap matrices, respectively, and E\mathbf{E} is the matrix of orbital energies.
The ground state energy EE of a system is a universal functional of the electron density ρ(r)\rho(\mathbf{r}): \nE[ρ]=T[ρ]+Vext[ρ]+Vee[ρ]E[\rho] = T[\rho] + V_{ext}[\rho] + V_{ee}[\rho]\nwhere T[ρ]T[\rho] is the kinetic energy functional, Vext[ρ]V_{ext}[\rho] is the external potential energy functional, and Vee[ρ]V_{ee}[\rho] is the electron-electron interaction functional. The Kohn-Sham equations are used to solve this functional:\n\left(-\frac{\hbar^2}{2m}\nabla^2 + V_{eff}(\mathbf{r})) \phi_i(\mathbf{r}) = \epsilon_i \phi_i(\mathbf{r})\nwith Veff(r)=Vext(r)+12VHartree(r)+VXC(r)V_{eff}(\mathbf{r}) = V_{ext}(\mathbf{r}) + \frac{1}{2}V_{Hartree}(\mathbf{r}) + V_{XC}(\mathbf{r}).
Principle

VSEPR Theory

For a central atom AA surrounded by nn electron pairs (bonding or lone pairs), the total potential energy VtotalV_{total} is modeled as the sum of pairwise repulsive interactions between these pairs, PiP_i and PjP_j, based on their spatial separation rijr_{ij} and the bond angle θijk\theta_{ijk}. A simplified model is:\nVtotal=∑i<jKij(1rij2+1rij3cos⁡2(θijk−θA−i,A−j))V_{total} = \sum_{i<j} K_{ij} \left(\frac{1}{r_{ij}^2} + \frac{1}{r_{ij}^3} \cos^2(\theta_{ijk} - \theta_{A-i, A-j})\right)\nwhere KijK_{ij} are empirical repulsion constants.
The molecular orbitals ϕi\phi_i are approximated as a linear combination of the atomic basis functions χμ\chi_{\mu} (LCAO approximation):\nϕi=∑μcμiχμ\phi_i = \sum_{\mu} c_{\mu i} \chi_{\mu}\nwhere cμic_{\mu i} are the coefficients determined by solving the secular equation derived from the Fock matrix F\mathbf{F}:\ndet⁡(F−SEI)=0\det(\mathbf{F} - \mathbf{S} \mathbf{E} \mathbf{I}) = 0\nHere, S\mathbf{S} is the overlap matrix, E\mathbf{E} is the energy matrix, and I\mathbf{I} is the identity matrix.