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Approximation and Limitation in Quantum Chemistry

In principle, quantum chemistry is the study to solve Schrödinger Equation. However, to obtain a precise analytical solution is not possible for many-electron atoms or molecules due to many-body problem. Therefore, in order to solve for an approximated solution, it would require two levels of approximation.

Born-Oppenheimer Approximation

The first approximation is the Born-Oppenheimer Approximation, which assumes the behaviors of electrons as in a field of frozen nuclei. With this approximation, we can approximate the Hamiltonian.

The exact expression for Hamiltonian is,

$$H_{exact}=T_{el}+V_{el}+T_{nuc-nuc}+V_{el-nuc}$$

where ${\textstyle H_{exact}}$ is the exact form of Hamiltonian, ${\textstyle T_{el}}$ is the kinetic energy of the electron, ${\textstyle V_{el}}$ is the potential energy of the electron, ${\textstyle T_{nuc-nuc}}$is the overall kinetic energy of the two nuclei and ${\textstyle V_{el-nuc}}$is the overall potential energy of the electron and the nucleus.

In a field of frozen nuclei, ${\textstyle T_{nuc-nuc}}$ will be set to 0 since the nuclei are not moving, and we thereby obtain the following form,

$$H_{Approx}=H_{el}=T_{el}+V_{el-el}+V_{el-nuc}$$

Using the form above, we can solve the electronic Schrödinger equation at successive, yet frozen, nuclear configurations.

An application of Born-Oppenheimer Approximation is to generate the potential energy (PE) curve, where potential energy is plotted as a function of internuclear distance, R. For a diatomic molecule (e.g. H₂), the PE curve generated by Born-Oppenheimer Approximation is shown as below,

PE curve for a diatomic molecule

Repulsive curve represents the repulsive force. Bound Curve describes the net effect of repulsive force and attractive force.

Orbital approximation

Orbital approximation is the second approximation. As we obtain the hermitian operator from Born-Oppenheimer approximation, where

$H_{el}=T_{el}+V_{el-el}+V_{el-nuc}$

We mentioned that ${\textstyle V_{el-el}}$ is not separable. Therefore, we approximate the wave function of the specific orbital in a Hartree product (hp).

The Hartree Product for the orbital approximation is ${\Psi }_{hp}={\psi }_{1}(1){\psi }_{2}(2)...{\psi }_{N}(N)$, where ${\psi }_{i}={\phi }_{i}{\sigma }_{i}$. ${\phi }_{i}$ represents the spatial orbital and ${\sigma }_{i}$represents the spin function.

If we ignore repulsion and parameterizing, the Hartree Product above can lead us to the  extended Hückel Theory and Tight Binding Approximation. These can be useful for extended systems.

However, if we want to recover electron repulsion, we can use the approximated orbital wave function and the corrected Hamiltonian, which are achieved by the Variational Principle, where the expectation value of the approximated Energy, ${\textstyle \langle E_{approx}\rangle }$, is solved in the following Dirac notation, ${\textstyle \langle {\Psi }_{hp}|H|{\Psi }_{hp}\rangle }$. This notation is equivalent to the integral,

${\textstyle \langle E_{approx}\rangle =\int _{0}^{all\,space}\psi _{hp}^{*}H_{el}\psi _{hp}d\tau }$

It is noteworthy that the exact energy is always less than the approximated energy. Therefore, ${\textstyle E_{approx}<E_{exact}}$.