17.4: Atomic Units
- Page ID
- 60581
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The electronic Hamiltonian is expressed, in this Section, in so-called atomic units (aus)
\[ H_e = \sum\limits \left[ -\dfrac{1}{2}\nabla_j^2 - \sum\limits_a\dfrac{Z_a}{r_{j,a}} \right] + \sum\limits_{j<k}\dfrac{1}{r_{r,k}}. \nonumber \]
These units are introduced to remove all \(\hbar \text{, e, and m}_e\) factors from the equations.
To effect this unit transformation, one notes that the kinetic energy operator scales as \(r_j^{-2}\) whereas the coulombic potentials scale as \(r_j^{-1}\) and as \(r_{j,k}^{-1}\). So, if each of the distances appearing in the cartesian coordinates of the electrons and nuclei were expressed as a unit of length \(a_0\) multiplied by a dimensionless length factor, the kinetic energy operator would involve terms of the form \(\left(-\frac{\hbar^2}{2(a_0)^2m_e} \right)\nabla_j^2 \), and the coulombic potentials would appear as \( \frac{Z_ae^2}{(a_0)r_{j,a}} \text{ and } \frac{e^2}{(a_0)r_{j,k}} \). A factor of \(\frac{e^2}{a_0}\) (which has units of energy since \(a_0\) has units of length) can then be removed from the coulombic and kinetic energies, after which the kinetic energy terms appear as \( -\frac{\hbar^2}{2(e^2a_0)m_e}\nabla_j^2 \) and the potential energies appear as \( \frac{Z_a}{r_{j,a}} \text{ and } \frac{1}{r_{k,j}}\). Then, choosing \( a_0 = \frac{\hbar^2}{e^2m_e} \) changes the kinetic energy terms into \( -\frac{1}{2}\nabla_j^2 \); as a result, the entire electronic Hamiltonian takes the form given above in which no \(e^2, \text{ me, or } \hbar^2\) factors appear. The value of the so-called Bohr radius \(a_0 = \frac{\hbar^2}{e^2m_e}\) is 0.529 Å, and the so-called Hartree energy unit \( \frac{e^2}{a_0} \), which factors out of \(H_e\), is 27.21 eV or 627.51 kcal/mol.