Data and formulas

$h =6.626\,068\,8\times 10^{-34}\,\mathrm{J/Hz}$

$\hbar = \displaystyle\frac{h}{2\pi} = 1.054\,571\,60\times 10^{-34}\,\mathrm{J\,s}$

$1\,\mathrm{amu} = 1.660\,538\,73\times 10^{-27}\,\mathrm{kg}$

$E = h\nu$

$$\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$$

$E^\mathrm{var} = \displaystyle\frac{\langle\phi\vert\hat{H}\vert\phi\rangle}{\langle\phi\vert\phi\rangle}$

The energy of a system perturbed by an additive Hamiltonian term $\hat{H}^{(1)}$ can be expressed in the form E = E⁽⁰⁾ + E⁽¹⁾, where

$$E^{(1)} = \langle\psi^{(0)}\vert\hat{H}^{(1)}\vert\psi^{(0)}\rangle.$$

If $\ell_1$ and $\ell_2$ are two angular momentum quantum numbers, the quantum number for the sum of the associated vectors can take on any of the following values:

$$\vert\ell_1-\ell_2\vert,\vert\ell_1-\ell_2\vert+1,\ldots,\ell_1+\ell_2-1,\ell_1+\ell_2.$$

If $\hat{A}$, $\hat{B}$ and $\hat{C}$ are arbitrary operators and k is a constant,

$$\begin{eqnarray*}[\hat{A},\hat{A}]& = & 0\\ \mbox{}[\hat{A},\hat{B}] & = & -[\h... ...t{C}] & = & \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B} \end{eqnarray*}$$

Additionally, $[\hat{x},\hat{p}_x] = i\hbar$.

Terms for ground states of p block elements:

Configuration	Terms
p or p5	$\isotope{2}{}{P}$
p2 or p4	$\isotope{3}{}{P}$, $\isotope{1}{}{D}$, $\isotope{1}{}{S}$
p3	$\isotope{4}{}{S}$, $\isotope{2}{}{D}$, $\isotope{2}{}{P}$

For the particle in a box,

$$\psi_n = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$

and

$$E_n = \frac{n^2h^2}{8mL^2},$$

with $n=1,2,3,\ldots$

In spherical polar coordinates,

$$dV = r^2\sin\theta\,dr\,d\theta\,d\phi.$$

The polar angles have the following ranges:

$$\begin{eqnarray*}\theta & \in & [0,\pi],\\ \phi & \in & [0,2\pi). \end{eqnarray*}$$

The rotational absorption selection rule is $\Delta J=1$. The rotational energy of a diatomic molecule is given by

$$E_J = J(J+1)\frac{\hbar^2}{2\mu R^2}$$

with $J = 0,1,2,\ldots$

To use ghost atoms in HyperChem: Select the atom you want to turn into a ghost atom. Choose Name Selection from the Select menu. Name the selected atom ``ghost-atoms''. Unselect the atom. Enable ghost atoms in the advanced options dialog of the ab initio setup window. Verify that the charge and multiplicity are correct in the options dialog.