Hartree-Fock Theory

Another ab initio method for solving the many-body Schrödinger equation (equation (3.5)), is Hartree-Fock theory. As discussed in section 3.2.3, this method attempts to transform the full -body equation into single-body equations. According to quantum mechanics, one can determine the ground state of the Hamiltonian, by means of the variational principle for the normalized wavefunction :

$$\left\langle \Psi \left\vert H\right\vert \Psi \right\rangle... ...N)H\Psi (1,2,\cdots ,N)dr_{1}dr_{2}\cdots dr_{N}=minimum=E_{0}$$ (3.14)

where is the spin direction of the th electron. According to this variational principle, an approximate wavefunction can be found which minimizes the expectation value , since the expectation value is always greater than the true ground state energy. Assuming that the electrons are independent, then is a Slater determinant of single-particle wavefunctions such that:

$$\Psi (1,2,\cdots N)=\frac{1}{\sqrt{N!}}\left\vert \begin{arr... ...& \psi _{2}(N) & \cdots & \psi _{N}(N) \end{array}\right\vert$$ (3.15)

where is the one-electron wavefunction of the th level, which depends on the position, , and spin direction, , of the th electron, and forms an orthonormal set:

$$\left\langle \psi _{\lambda }\mid \psi _{\nu }\right\rangle ... ...{*}(i)\psi _{\nu }^{}(i)d\mathbf{r}_{i}=\delta _{\lambda \nu }$$ (3.16)

Using equation (3.15) as a trial solution, the expectation value can be evaluated as:

$$\left\langle \Psi \left\vert H\right\vert \Psi \right\rangle... ...t\vert U\right\vert \psi _{\nu }\psi _{\lambda }\right\rangle$$ (3.17)

where the Hamiltonian is divided into the one-electron part and the electron-electron Coulomb interaction U. represents the 1st and 3rd terms of equation (3.6) and U the remaining term. Using Lagrangian multipliers to keep the normalization condition, equation (3.16), the variational problem of equation (3.17) is solved:

$$\sum _{\lambda =1}^{N}\left\langle \delta \psi _{\lambda }\l... ...e \delta \psi _{\lambda }\mid \psi _{\lambda }\right\rangle =0$$ (3.18)

where denotes the Lagrangian multiplier. To satisfy equation (3.18) for an arbitrary variation , the one-electron wavefunction should satisfy:

$$H_{0}\psi _{\lambda }(i)+\left[ \sum _{\nu =1}^{N}\sum _{s_{... ...right] \psi _{\nu }(i)=\epsilon _{\lambda }\psi _{\lambda }(i)$$ (3.19)

Equation (3.19) is known as the Hartree-Fock equation and the use of the single Slater determinant, equation (3.15), to express the many-electron wavefunction is known as the Hartree-Fock approximation. The Hartree-Fock equation represents the one-electron approximation for interacting fermions which includes the anti-symmetry of the wavefunction or exchange interaction. If this contribution is ignored, then it is called the Hartree approximation.