Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Molecular Properties

The SCF method provides a computationally tractable means of obtaining an approximate wave function and absolute, total energy for a many electron system; however, as mentioned in the introduction, the goal of electronic structure theory is to provide a method of predicting a large array of chemical and molecular properties, not just energies and wave functions. Many of the properties of interest can be found by constructing the full potential energy surface (PES) by performing an SCF calculation on a wide variety of nuclear configurations. From the PES one obtains the potential for nuclear motion under the BO approximation, which allows one to solve the nuclear wave equation for rotational and vibrational frequencies. By finding the minima on the PES, one can find stable nuclear configurations that could be observed in the laboratory and report spectroscopic information such as bond lengths, bond angles, and rotational constants to experimentalists. One can also find other stationary points on the PES which correspond to transition states for interconversion between minima and, thereby, provide experimentalists with reaction pathways and energy barriers. The amount of data one can extract from the PES is enormous, but the calculation of an SCF energy for a large number of nuclear configurations is often too time consuming a task. In order to reduce the computational effort and yet extract as much of the desired information as possible, quantum chemists use derivatives of the absolute energy. [21] From the gradient of the energy one can find stationary points such as minima and transition states by following paths of steepest ascent or descent. From the second derivative one can find the curvature of the PES around a stationary point and predict harmonic vibrational frequencies. From higher derivatives one can find higher order spectroscopic constants such as anharmonisity. One can find reasonable descriptions of these properties in almost any spectroscopy book. [22,23,24]

In addition to properties that can be found by examining the PES or by taking energy differences, there are a number of properties which depend on the distributions of the electrons as described by the wave function. In order to examine how one may extract such information, consider the calculation of a one-electron property such as the dipole moment. For a one-electron property the requisite operator takes the general form

$\begin{displaymath}\hat{O} = \sum_i^N \hat{o}_i, \end{displaymath}$

(83)

and the expectation value of $\hat{O}$ as given by Slater's rules is

$\begin{displaymath}\langle \Psi_0 \vert \hat{O} \vert \Psi_0 \rangle = \sum_a^{\... ...t \hat{o} \vert \psi_a) = \sum_{uv} P_{uv} (v\vert o\vert u), \end{displaymath}$

(84)

where P_uv is an element of the density matrix. For example, the classical definition of the dipole moment of a collection of charges q_i with position vectors ${\bf r_i}$ is given by the sum

$\begin{displaymath}\mu = \sum_i q_i {\bf r_i}. \end{displaymath}$

(85)

In quantum mechanics, the corresponding definition of the dipole moment of a molecule is

$\begin{displaymath}\mu = \langle \Psi_0 \vert -\sum_i^N {\bf r_i} \vert \Psi_0 \rangle = \sum_A Z_A {\bf R_A}, \end{displaymath}$

(86)

where the contributions of the electrons and nuclei have been separated into the first and second sums, respectively. From equation (1.85), the electronic contribution can be rewritten in terms of the known atomic orbital basis and the density matrix producing

$\begin{displaymath}\mu = -\sum_{uv} P_{uv} (v\vert{\bf r}\vert u) + \sum_A Z_A {\bf R_A}. \end{displaymath}$

(87)

With the exception of the $(v\vert{\bf r}\vert u)$ integrals which must be computed in a separate step, all of the required information is available from the SCF wave function. In conclusion, one of the most important one-electron properties which can be computed in this manner is the distribution of the electrons themselves. The charge density, which represents the probability of finding an electron in a specified region of space, is given by

$\begin{displaymath}\rho({\bf r}) = \sum_{uv} P_{uv} \phi_u ({\bf r}) \phi_v^* ({\bf r}). \end{displaymath}$

(88)

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