14.1: Time-Dependent Vector Potentials

The full N-electron non-relativistic Hamiltonian H discussed earlier in this text involves the kinetic energies of the electrons and of the nuclei and the mutual Coulombic interactions among these particles

\[H = \sum\limits_{a=1,M} -\left(\dfrac{\hbar^2}{2m_a}\right) \nabla_a^2 + \sum\limits_j \left[ \left(-\dfrac{\hbar^2}{2m_e}\right) \nabla_j^2 - \sum\limits_a Z_a \dfrac{e^2}{r_{j,a}} \right] + \sum\limits_{j<k}\dfrac{e^2}{r_{j,k}} + \sum\limits_{a<b}Z_aZ_b \dfrac{e^2}{R_{a,b}}. \nonumber \]

When an electromagnetic field is present, this is not the correct Hamiltonian, but it can be modified straightforwardly to obtain the proper H.

The Time-Dependent Vector \(\textbf{A}(\textbf{r},t)\) Potential

The only changes required to achieve the Hamiltonian that describes the same system in the presence of an electromagnetic field are to replace the momentum operators P\(_a\) and p\(_j\) for the nuclei and electrons, respectively, by (P\(_a\) - Z\(_a\) e/c A(R\(_a\),t)) and (p\(_j\) - e/c A(rj ,t)). Here Za e is the charge on the ath nucleus, -e is the charge of the electron, and c is the speed of light.

The vector potential A depends on time t and on the spatial location r of the particle in the following manner:

\[ \textbf{A}(\textbf{r},t) = 2 \textbf{A}_0 cos(\omega t - \textbf{k}\cdot{\textbf{r}}). \nonumber \]

The circular frequency of the radiation \(\omega\) (radians per second) and the wave vector k (the magnitude of k is |k| = \(\frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength of the light) control the temporal and spatial oscillations of the photons. The vector \(\textbf{A}_o\) characterizes the strength (through the magnitude of \(\textbf{A}_o\)) of the field as well as the direction of the A potential; the direction of propagation of the photons is given by the unit vector k/|k|. The factor of 2 in the definition of A allows one to think of \(\textbf{A}_0\) as measuring the strength of both \( e^{i(\omega t - \textbf{k}\cdot{\textbf{r}})} \) and \( e^{i(\omega t - \textbf{k}\cdot{\textbf{r}})} \) components of the \( cos(\omega t - \textbf{k}\cdot{\textbf{r}}) \) function.

The Electric \(\textbf{E}(\textbf{r},t) \text{ and Magnetic } \textbf{H}(\textbf{r},t) \text{ Fields }\)

The electric \(\textbf{E}(\textbf{r},t) \text{ and magnetic } \textbf{H}(\textbf{r}\),t) fields of the photons are expressed in terms of the vector potential A as

\[ \textbf{E}(\textbf{r},t) = -\dfrac{1}{3}\dfrac{\partial \textbf{A}}{\partial t} = \dfrac{\omega}{c}\textbf{A}_0 \sin( \omega t - \textbf{k}\cdot{\textbf{r}} ) \nonumber \]

\[ \textbf{H}(\textbf{r},t) = \nabla \textbf{ x A } = \textbf{ k x A}_o 2 \sin(\omega t - \textbf{k}\cdot{\textbf{r}}). \nonumber \]

The E field lies parallel to the \(\textbf{A}_o\) vector, and the H field is perpendicular to \(\textbf{A}_o\); both are perpendicular to the direction of propagation of the light k/|k|. E and H have the same phase because they both vary with time and spatial location as \(\sin (\omega t - \textbf{k}\cdot{\textbf{r}}).\) The relative orientations of these vectors are shown below.

Figure 14.1.1: Insert caption here!

The Resulting Hamiltonian

Replacing the nuclear and electronic momenta by the modifications shown above in the kinetic energy terms of the full electronic and nuclear-motion hamiltonian results in the following additional factors appearing in H:

\[ H_{int} = \sum\limits_j \left[ \dfrac{ie\hbar}{m_ec}\textbf{A}(r_j,t)\cdot{\nabla_j} + \left( \dfrac{e^2}{2m_ec^2} \right)|\textbf{A}(r_j,t)|^2 \right] + \sum\limits_a \left[ \left( iZ_a\dfrac{e\hbar}{m_ac} \right)\textbf{A}(R_a,t)\cdot{\nabla_a} + \left( \dfrac{Z_a^2e^2}{2m_ac^2} \right)|\textbf{A}(R_a,t)|^2 \right]. \nonumber \]

These so-called interaction perturbations \(H_{int}\) are what induces transitions among the various electronic/vibrational/rotational states of a molecule. The one-electron additive nature of \(H_{int}\) plays an important role in determining the kind of transitions that \(H_{int}\) can induce. For example, it causes the most intense electronic transitions to involve excitation of a single electron from one orbital to another (e.g., the Slater-Condon rules).