Molecular States

Abstract

This chapter will introduce the quantum mechanical equation of motion, i.e., the time-dependent Schrödinger equation. We will show how the separation of variables can be exploited to partition the molecular wavefunction in translational, rotational, vibrational, and electronic components, with special emphasis on the Born–Oppenheimer approximation and its breakdown. We shall then provide an overview of the electronic structure and reactivity of excited states commonly found in organic molecules.

Problems

2.1

Consider a one dimensional system described by the following Hamiltonian

$$ \hat{H}(x) = -\frac{\hbar ^2}{2m}\frac{{\mathrm{d}}^2}{{\mathrm{d}}x^2} - F x \qquad (F > 0) $$

appropriate, for example, for a particle of mass m and charge q in the presence of a constant electric field $E_0$ (in that case $F=q E_0$). Working in the momentum representation (use $x={\mathrm{i}}\hbar {\mathrm{d}}/{\mathrm{d}}p$), evaluate the time evolution of a Gaussian wavepacket, i.e., find $\varPsi (p,t)$ given $\varPsi (p,0) = (2\alpha /\pi )^{-1/4}\exp (-\alpha p^2)$, with $\alpha > 0$.

2.2

The potential energy of a Morse oscillator is $V(x) = D(1-{\mathrm{e}}^{-a(x-x_0)})^2$, where D is the well depth. The corresponding energy eigenvalues are

$$ E_v = \hbar \omega (v+1/2) + \frac{\hbar ^2\omega ^2}{4 D} (v+1/2)^2 \qquad \omega = a\sqrt{2D/m} $$

where $v = 0,1,\ldots$ and m is the mass of the oscillator. Evaluate the density of states for a single Morse oscillator and use it to obtain a classical expression (i.e., without taking into account the quantization of energy) for the density of states of two noninteracting identical Morse oscillators.

2.3

In molecules where $S_1$ and $T_1$ are close in energy, the “inverse” ISC from $T_1$ to $S_1$ may be characterized by a non-negligible rate constant $K_{invISC}$. Compute $K_{invISC}$ for acetone, according to the following data/assumptions: (i) the molecule is isolated, so that its energy is constant, and in particular $E= 1$ eV above the ZPE of $T_1$; (ii) the lifetimes of $S_1$ and $T_1$ are long enough to reach microcanonical equilibrium before decaying to $S_0$; (iii) the adiabatic (i.e., minimum-minimum) energy difference between $S_1$ and $T_1$ is $\varDelta E=0.25$ eV; (iv) $S_1$ and $T_1$ have the same vibrational frequencies; (v) the rate constant for the ISC $S_1 \rightarrow T_1$ is $K_{ISC} = 3.5$ ns$^{-1}$.

2.4

Four different Slater determinants can be written with two electrons in two orbitals. The related matrix elements of $H_{el}$ are

$$ \begin{aligned} \left\langle \phi _i\wedge \overline{\phi }_i \left| \hat{H}_{el} \right| \phi _i\wedge \overline{\phi }_i \right\rangle&= 2\varepsilon _i + J_{ii} \\ \left\langle \phi _i\wedge \overline{\phi }_j \left| \hat{H}_{el} \right| \phi _i\wedge \overline{\phi }_j \right\rangle&= \varepsilon _i + \varepsilon _j + J_{ij} \\ \left\langle \overline{\phi _i}\wedge \phi _j \left| \hat{H}_{el} \right| \phi _i\wedge \overline{\phi }_j \right\rangle&= -K_{ij} \end{aligned} $$

with $i,j=1,2$ and where $\phi _i$ (respectively, $\overline{\phi }_i$) label a spin-orbital with spin part $\alpha$ (respectively, $\beta$). $\varepsilon _i$ are the orbital energies ($\varepsilon _1 \le \varepsilon _2$). $J_{ij}$ are the Coulomb integrals, representing the electrostatic repulsion between the two charge clouds $\left| \phi _i\right| ^2$ and $\left| \phi _j\right| ^2$. $K_{12}$ is the exchange integral, see Eq. (2.93). Both the Coulomb and the exchange integrals are positive quantities and we assume $J_{ii} > J_{ij}$. Write $\varphi _{S_0}$, $\varphi _{S_1}$ and $\varphi _{T_1}$ in terms of the three Slater determinants $\phi _1\wedge \overline{\phi }_1$, $\phi _1\wedge \overline{\phi }_2$, and $\overline{\phi }_1\wedge \phi _2$. Evaluate the corresponding energies. Which one is the ground state when the two orbitals are degenerate (i.e., $\varepsilon _1=\varepsilon _2$)?