Introduction to Electronic structure Reading notes: Chapter II-Multiple electron wave functions and operators

Updated: 3 May 2016

General notation for multi-electron Hamiltonian operators:

\ (\mathscr{h}=-\sum\limits_{i=1}^{n}\dfrac{1}{2}\nabla_i^2-\sum\limits_{a=1}^{m}\dfrac{1}{2m_a}\nabla_a^2-\sum \limits_{i=1}^{n}\sum\limits_{a=1}^{m}\dfrac{z_a}{r_{ia}}+\sum\limits_{i=1}^{n}\sum\limits_{j>i}^{n}\dfrac{ 1}{r_{ij}}+\sum\limits_{a=1}^{m}\sum\limits_{b>a}^{m}\dfrac{z_az_b}{r_{ab}}\)

Note: 1. Adopt the atomic unit of units; 2. The relativistic effect was not considered; 3. No outfield

2.1 Description Electronics

2.1.1 Atomic units (a.u.):

Length unit: Bohr \ (=\dfrac{4\pi\varepsilon_0\hbar^2}{m_ee^2}=a_0=0.52918\ \overset{\circ}{\rm{a}}\)

Energy unit: Hartree \ (=\dfrac{e^2}{4\pi\varepsilon_0a_0}=\mathscr{e}_a=27.211\ \rm{ev}=627.51\ \rm{kcal/mol}\)

Quality Unit: Electronic quality \ (m_e=9.1095\times10^{-31}\ \rm{kg}\)

Charge unit: Electronic charge amount \ (e=1.6022 \times 10^{-19}\ \rm{c}\)

Angular momentum Unit: The approximate Planck constant \ (\hbar=1.0546\times 10^{-34}\ \rm{j\cdot s}\)

2.1.2 Bohr-oppenheimer approximation

Principle of 2.1.3 Electron wave function anti-weigh, spin and bubble-profit incompatibility

2.2 Description Track

2.2.1 Spin orbit and space orbit

The electron wave function considering the electron spin and space distribution is called the spin orbit spin orbitals, and the electron wave function which only considers the spatial distribution becomes spatial orbit spatial Orbital.

2.2.2 Hartree Product

2.2.3-Slater determinant

Line: The same atom occupies a different spin orbit; column: The same spin track places different atoms. n electrons occupy n spin orbits.

Coefficient: \ ((n!) ^{-\frac{1}{2}}\)

is to make the Hartree product satisfy the linear combination of anti-symmetry.

2.2.4 Hartree-fock approximation

Fock operator: \ (f (i) =-\DFRAC{1}{2}\NABLA_I^2-\SUM\LIMITS_{A=1}^{M}\DFRAC{Z_A}{R_{IA}}+V^{HF} (i) \)

Hartree-fock equation: \ (f (i) \chi (\textbf{x}_i) =\varepsilon\chi (\textbf{x}_i) \)

The key is the single electron potential energy term \ (V^{HF} (i) \) for the average potential energy of other electrons to the first electron. See chapter below for specific definitions.

Self-consistent field method SCF: Guesses the initial spin orbit, thus the mean field is calculated by the Coulomb's law, and by the mean field substituting Fock operator, a new set of ground-state spin orbits are obtained by means of the variational method. Repeat this process until the change in energy and orbit is less than the error range.

The difficulty: Each hartree-fock equation can solve a set of intrinsic values of the first electron and the intrinsic function (infinity) of each other, and in the form of all electronic Fock operator forms the same, meaning that n electrons will occupy the same infinite number of spin orbits.

The Rooothaan equation is described in detail in the following chapter.

2.2.5 Example: Minimum base H2 model

Overlapping integrals s, exchanging integrals

2.2.6 Excitation State determinant

Single-excitation, double-excitation ... The System wave function (the Slater determinant of the Dirac notation) is replaced by the spin orbit of the ground state in the original empty spin orbit.

2.2.7 exact wave function and configuration interaction

Ideas:

Set \ (\{\chi_i (x) \}\) is a set of complete bases (infinitely multiple elements) solved above. Because of the approximate limitation of hartree-fock, no filling method can accurately represent the state of the system. But the state of the system can be expressed as a linear superposition of various filling methods (System wave function, Slater determinant), i.e.

\ (|\phi\rangle=c_0|\psi_0\rangle+\sum\limits_{ra}|\psi_a^r\rangle+\sum\limits_{a<b \atop r<s}c_{ab}^{rs}|\ Psi_{ab}^{rs}\rangle+\sum\limits_{a<b<c \atop r<s<t}c_{abc}^{rst}|\psi_{abc}^{rst}\rangle+\cdots\)

Introduction to Electronic structure Reading notes: Chapter II-Multiple electron wave functions and operators