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Quantum chemistry influences all  branches of chemistry by applying quantum mechanic to chemistry problems. It can be considered a subarea of physical chemistry. In brief, physical chemistry studies the macroscopic, atomic and particular phenomena of the chemical systems. And in particular, the quantum chemistry part of it adopts the concept of quantum mechanics to understand, for example, molecular states and properties (for example bond length and rotation), and thermodynamic properties (for example heat capacity, entropy).

Organic chemistry adopts quantum mechanics to investigate mechanisms of organic reactions, to estimate the stability of molecules, and to analyze NMR spectrums. Analytical chemistry uses quantum mechanics to determine the frequency and intensity in the spectroscopy. Inorganic chemistry approximates quantum mechanical methods and properties of transitional-metal ion with ligand field theory. Biochemical method uses quantum chemistry for bio-molecular calculation.

The Early Days of Quantum Chemistry
The age of Quantum Chemistry began in the 1910s under two traditions: physical chemistry and molecular spectroscopy. Gilbert Newton Lewis, and Niels Bohr were among the first scientists in this field of study.

On April 5th, 1913, Bohr published “On the Constitution of Atoms and Molecules”, which was considered to be one of the first article in Quantum Chemistry. In this article, he used a system of several nuclei with electrons surrounding it in a way as if there weren’t any other nuclei. The electrons orbit in the circle connected to the nuclei and this ring was the only thing that connected the system together. However, there was a small number of electrons in the outer levels were arranged differently. Along with that, Bohr also suggested that the hydrogen molecule offered the heat of formation, which was double compared to the experimental data. For this reason, the calculations were too complicated for more complex molecules.

In 1926, Lewis suggested the model in which the atom shared electron pairs satisfied even complicated molecules. For the simplest molecules, such as hydrogen, the electrons were classified into two different groups. The first group suggested that the electron orbited around both nuclei, while the other one suggested was similar to that of Bohr’s model, in which the electron move either in a plane either perpendicular to the axis of the two nuclei or in the cross orbits.

Beside Bohr and Lewis, Niels Bjerrum, a friend of Bohr, whom followed his classical electro dynamical identity between the radiation frequency and motion one. This model was called the hybrid model, which quantified the rational energy. With this model, Bjerrum suggested a model of the infrared molecular spectra for simple diatomic molecule and established the base for other quantum theory.

Then in 1919 – 1920, three scientists Torsten Herrlinger, Adolf Kratzer, and Karl Schwarzschild began to expand Bohr’s work. According to their work, the motion of electron could be applied into the interpretation of the rational and vibrational motions of molecule under a theoretical umbrella. Edwin Crawford Kemble, an American physicist indicated the interpretation of band spectra that spectroscopic frequencies are the energy differences and are not the same as the ones of motion like they were mentioned in Bjerrum’s theory.

Heitler and London
Walter Heitler and Fritz London were the first scientists to be successful in providing a proper structure for the hydrogen molecule. In the early 1927, they both traveled to the University of Zurich to study with Erwin Schrodinger, whom developed one of the most important equations in quantum mechanic under his name.

In the beginning, Heitler and London’s goal was to determine the interaction of the charges of two atoms. However, the Coulomb integral, which represent electron’s energy in an atomic orbital, was not supportive to their calculation. Even with help from Heisenberg’s work regarding the quantum mechanical resonance phenomenon, Heitler and London were not able to achieve their goal.

Years later, Heitler and London were still trying to figure the problem out and began to perform their calculation by considering the two hydrogen atom approaching closely to one another. They made an assumption that the two electron 1 and 2 belonged to two different atoms a and b respectively. Because the two electrons were identical, the wave function was written as:

$$\Psi=c_1\Psi_a(1)\Psi_b(2)+c_2\Psi_a(2)\Psi_b(1)$$

And they also minimized the energy in other to calculate for  and :

$$E=\frac{\int \Psi H\Psi d\Tau}{\int \Psi^2d\Tau}$$

They ended up with the two results for energy:

$$E_1=2E_0+\frac{C+A}{1+S_{1,2}} $$ and $$E_2=2E_0+\frac{C-A}{1-S_{1,s}}$$

Where S is the overlap of two atomic wave functions ; C is the Coulomb integral, and A is the exchange integral, where both C and A are negatives. This indicates that $$\frac{c_1}{c_2}=1$$ for $$E_1$$ and $$\frac{c_1}{c_2}=-1$$ for $$E_2$$,

The wave functions were rewritten as:

$$\Psi_1=\Psi_a(1)\Psi_b(2)+\Psi_a(2)\Psi_b(1)$$

$$\Psi_2=\Psi_a(1)\Psi_b(2)-\Psi_a(2)\Psi_b(1)$$

Density functional theory
Density functional theory (DFT) is one of the computational quantum mechanical methods used in quantum chemistry. It uses the modeling method to investigate the many-body-system electronic structure s in particular atoms, molecules, and the condensed phases. The pioneer DFT model, Thomas–Fermi model, was developed independently by Thomas and Fermi in 1927. This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave function, although not very successful in the treatment of entire molecules. The method did provide the basis for what is now known as density functional theory, which is given by,

$$T_{tf}(n)=C_F\int n(\overrightarrow{r})n^{(2/3)}(r)d^3r =C_F\int n^{5/3}(\overrightarrow{r})d^3r. $$

Hartree-Fock method came later as a method to approximate the wave function and the energy of a many-body system in the stationery state. It majorly made the band energy calculation practical for crystalline solids, surfaces and molecules. Density functional in quantum chemistry was then lead by Johnson and his coworkers in the Slater’s group in 1960 with the invention of finite system Scatter wave computer system to simplify the Hartree-Fock method. The system assumes an approximation with a single Slater determinant, in the cases of fermion particles, or with a single bosons particle permanent of N orbitals.

Kohn and Sham laid the basis of modern DFT method by proving that electron density could be used as the fundamental property to develop the many-body system. The important difference from previous method was that the equations were no more approximation of Hartree-Fock method as in Slater equation, but an original theory in the same logical base level as Hartree-Fock method. In contrast to the traditional approximation method, modern DFT systematically treats the many-body system with $$ \hat U $$ to be a single-body system without $$ \hat U $$. The normalized $$\,\!\Psi$$ is adopts the electronic density $$n(\vec r)$$ and is given  by,

$$n(\vec r) = N \int{\rm d}^3r_2 \cdots \int{\rm d}^3r_N \Psi^*(\vec r,\vec r_2,\dots,\vec r_N) \Psi(\vec r,\vec r_2,\dots,\vec r_N).$$

Density functional is split into four terms; the Kohn-Sham kinetic energy, an external potential, exchange and correlation energies. A large part of the focus on developing DFT is on improving the exchange and correlation terms. Though this method is less developed than post Hartree–Fock methods, its significantly lower computational requirements (scaling typically no worse than basis functions, for the pure functionals) allow it to tackle larger polyatomic molecules and even macromolecules. This computational affordability and often comparable accuracy to MP2 and CCSD(T) (post-Hartree–Fock methods) has made it one of the most popular methods in computational chemistry at present.

Many-Electrons Model
Helium is the first multi-electron system from the period table. The helium atom, a relatively simple quantum system, can be used to illustrate the techniques used for more complex atoms. The atomic unit, a system of units, is commonly used in atomic and molecular calculations to simplify the equations. A natural unit of angular momentum on an atomic or molecular scale is ħ. A natural unit of length on an atomic scale is the Bohr radius:

$$a_0=\frac{4\pi\varepsilon_0\hbar}{m_0e^2}$$

A natural unit of energy is

$$E=\frac{m_ee^4}{\pi^2\varepsilon_0^2\hbar^2}$$

Using natural units of energy and length, the atomic unit of energy $$E_h$$ can be derived as

$$E_h=\frac{m_ee^4}{16\pi^2\varepsilon_0^2\hbar^2}$$

The use of atomic unit greatly simplifies most of the equations for atomic and molecular calculations. This can be applied to the Hamiltonian operator of a helium atom in atomic units in order to calculate energy independent of the physical constants.

$$\hat{H}=-\frac{\hbar}{2m_e}\nabla_1^2-\frac{\hbar}{2m_e}\nabla_2^2-\frac{2e^2}{4\pi\epsilon_0r_2}+\frac{e^2}{4\pi\epsilon_0r_{12}}$$

$$\hat{H}=-\frac{1}{2}\nabla_1^2-\frac{1}{2}\nabla_2^2-\frac{2}{r_1}-\frac{2}{r_2}+\frac{1}{r_{12}}$$

By using atomic units, the calculated energies change as the value for the constants change.

Calculating Energy with Hartree-Fock Method

The energies can be calculated in many ways. Primarily the exact energy of an atom is taken to be the value experimentally found. One method to find the energies is through the Hartree-Fock Method. Hartree-Fock procedure for a helium atom is to write two-electron wave function as a product of orbitals.

$$\psi(r_1,r_2)=\psi(r_1)\psi(r_2)$$

According to the Pauli Exclusion Principle, it is assumed that both electrons are in the same orbital. The two functions on the right side are the following equation. The superscript “eff” emphasizes that it is an effective, or average, potential.

$$V_1^{eff}(r_1)=\int \psi^*(r_2)\frac{1}{r_{12}}\psi(r_2)dr_2$$

The effective one electron Hamiltonian operator is defined as

$$\hat{H_1}^{eff}(r_1)=-\frac{1}{2}\nabla_1^2-\frac{2}{r_1}+{V_1}^{eff}(r_1)$$

With the effective one electron Hamiltonian operator, the Schrodinger equation becomes:

$$\hat{H_1}^{eff}(r_1)\psi(r_1)=\varepsilon_1\psi(r_1)$$

Many-Electron Problem
Quantum mechanics relies on solving the Schrodinger Equation for any quantum system to find the quantized energy levels, ground state, excited state, and other characteristics such as the wavefunctions that describe the motion of particles involved. Systems such as a single particle in an infinite well are described with differential equations that could be solved quite easily. However, the Schrodinger’s equation contains limitations, as it cannot be solved easily for the many-electrons systems due to the presence of many more interactions for which the equation needs to account.

Therefore, the Born-Oppenheimer Approximation is often used to simply the equation by splitting the wavefunction into two components, one for the nuclear motion and the other for the electronic motion. It is assumed that the nuclear motion is so slow that the nucleus is considered frozen, and therefore the wavefuncitons depend only on positions and not velocities. The Bohr-Oppenheimer Approximation does the following simplification:
 * Ψmolecule=ψelectronsψNucleus

Using this approximation, the many-electrons time-independent Schrodinger equation becomes,

$$H\Psi = \sum_{i=1}^{N}(\frac{-h^{2}}{2m}\bigtriangledown_{i}^{2}-\sum_{\alpha }{} \frac{Z_{\alpha}e^{2}}{4\pi\epsilon\mid r_{i}-d_{\alpha}\mid})\Psi + \frac{1}{2} \sum_{i} \sum_{j\neq i} \frac{e^{2}}{4\pi \epsilon \mid r_{i}-r_{j}\mid }\Psi = E\Psi$$

This equation is impossible to solve. Consequently, extensive research is conducted to construct experimental solutions based on trial and errors that explain the physical and chemical phenomena of such systems.

Research by P. Christov in Sofia University, Bulgaria, demonstrates the difficulty of such solutions and an attempt at finding solutions, which are similar to the behavior of a many-electron system.

The time-dependent quantum Monte Carlo (TDQMC) methode is employed to generate wavefunctions and ground state energy, which resemble the prediction made by standard diffusion Monte Carlo method.

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

In a field of frozen nuclei, $T_{nuc-nuc}$ 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 ,

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 $V_{el-el}$ 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, $\langle E_{approx}\rangle$, is solved in the following Dirac notation, $\langle {\Psi}_{hp}|H|{\Psi}_{hp}\rangle$. This notation is equivalent to the integral,

$\langle E_{approx}\rangle = \int_{0}^{all\, space} \psi_{hp}^*H_{el}\psi_{hp}d\tau$

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