Electronic molecular Hamiltonian

The electronic molecular Hamiltonian is the term of the molecular Hamiltonian obtained when the molecular geometry is frozen. Within the Born-Oppenheimer approximation, the electronic Hamiltonian is said to depend adiabatically on the molecular geometry. Its discrete eigenvalues are called potential energy surfaces and the corresponding eigenstates the electronic states of the molecule. The electronic states are labelled according to their group representation and spin multiplicity.

Related Topics:
Hamiltonian - Molecular Hamiltonian - Molecular geometry - Born-Oppenheimer approximation - Adiabatically - Eigenvalue - Potential energy surface - Eigenstate - Labelled - Group representation - Spin multiplicity

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The electronic Hamiltonian for a multi-electron molecule in atomic units is:

Related Topics:
Molecule - Atomic units

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hat H_{mathit el}=sum_{i}{-rac{1}{2} abla_i^2}-sum_{i}{sum_{a}{rac{Z_a}{left |mathbf{r_i}-mathbf{d_a} ight |}}}+rac{1}{2}sum_{i}{sum_{j e i}{rac{1}{left |mathbf{r_i}-mathbf{r_j} ight |}}}+rac{1}{2}sum_{a}{sum_{b e a}{rac{Z_a Z_b}{left |mathbf{d_a}-mathbf{d_b} ight |}}}

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where

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• $mathbf\{r\_i\}$ is the vector position of electron $i$ with vector components in Bohr radii,
• $Z\_a$ is the charge of fixed nucleus a in units of the elementary charge,
• $mathbf\{d\_a\}$ is the vector position of nucleus $a$ with vector components in Bohr radii.

The first term in the Hamiltonian is the sum of the kinetic energy operators for each electron in the system. The second term is the sum of the electron-nucleus Coulombic attractions. The third term is the sum of the electron-electron Coulombic repulsions. The final term is the sum of the nucleus-nucleus Coulombic repulsions, also known as the nuclear repulsion energy (see electric potential).

Related Topics:
Kinetic energy - Operator - Electric potential

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