《Quantum chemistry》章末总结10
本章从量子力学的角度考察化学键的组成，比上一章更“化学”一些。
Chapter 10 The Chemical Bond: One- and Two-Electron Molecules
10.1 The Born-Oppenheimer Approximation Simplifies the Schrödinger Equation for Molecules
For simplicity, let’s consier the simplest neutral molecule, $H_2$. The molecular Schrödinger equation for $H_2$ is
$$
\hat H_{mol} \psi_{mol}(\bold r_1, \bold r_2, \bold R_A, \bold R_B) = E_{mol} \psi_{mol}(\bold r_1, \bold r_2, \bold R_A, \bold R_B)
$$
where $\hat H_{mol}$ is given by
$$
\begin{aligned} \hat H_{mol} = &-\frac {\hbar^2}{2M} (\nabla_A^2 + \nabla_B^2) - \frac {\hbar^2}{2m_e}(\nabla_1^2 + \nabla_2^2) - \frac {e^2}{4\pi \epsilon_0 r_{1A}} \[3mm] &- \frac {e^2}{4\pi \epsilon_0 r_{1B}} - \frac {e^2}{4\pi \epsilon_0 r_{2A}} - \frac {e^2}{4\pi \epsilon_0 r_{2B}} + \frac {e^2}{4\pi \epsilon_0 r_{12}} + \frac {e^2}{4\pi \epsilon_0 R} \end{aligned}
$$
where $R=|\bold R_A - \bold R_B|$. $M$ is the mass of each hydrogen nucleus, $m_e$ is the mass of an electron, the subscripts A and B refer to the nuclei of the individual atoms, the subscripts 1 and 2 refer to the individual electrons.
Because the nuclei are so much more massive than the electrons, they move slowly compared to the electrons, and so the electrons adjust essentially instantaneously to any motion of the nuclei. A systematic treatment of this observation assumes that
$$
\psi_{mol}(\bold r_1, \bold r_2, \bold R_A, \bold R_B) \approx \psi_{el}(\bold r_1, \bold r_2, R) \psi_{nucl}(\bold R_A, \bold R_B)
$$
Then we obtain
$$
\hat H_{el} \psi_{el}(\bold r_1, \bold r_2, R) = E_{el}\psi_{el}(\bold r_1, \bold r_2, R)
$$
where
$$
\begin{aligned} \hat H_{el} = &- \frac {\hbar^2}{2m_e}(\nabla_1^2 + \nabla_2^2) - \frac {e^2}{4\pi \epsilon_0 r_{1A}} - \frac {e^2}{4\pi \epsilon_0 r_{1B}} \[3mm] &- \frac {e^2}{4\pi \epsilon_0 r_{2A}} - \frac {e^2}{4\pi \epsilon_0 r_{2B}} + \frac {e^2}{4\pi \epsilon_0 r_{12}} + \frac {e^2}{4\pi \epsilon_0 R} \end{aligned}
$$
In atomic units, the equation becomes
$$
\hat H_{el} = -\frac 12 (\nabla_1^2 + \nabla_2^2) - \frac {1}{r_{1A}} - \frac {1}{r_{1B}} - \frac {1}{r_{2A}} - \frac {1}{r_{2B}} + \frac {1}{r_{12}} + \frac {1}{R}
$$
We can obtain another equation for $\psi_{nucl}(\bold R_A, \bold R_B)$, represents the motion of the nuclei and leads to vibrational and rotational motion, which we have already discussed in Chapters 5 and 6.
The approximate separation is called the Born-Oppenheimer approximation. The essence of the Born-Oppenheimer is assuming that the complete molecular wave function can be factored into an electronic part and a nuclear part.
10.2 The Hydrogen Molecular Ion, $H_2^+$, Is the Prototype Diatomic Molecule
The hydrogen molecular ion, $H_2^+$, is a stable molecular species with an equilibrium bond length of 105.7 pm (1.997 $a_0$) and a potential-well depth, $D_e$, of 0.10264 $E_h$ (269.5 $KJ \cdot mol^{-1}$).
![QC-fig10.1]
The hydrogen molecular ion to some extent plays the same role for molecular calculations as the hydrogen atom does for atomic calculations. we shall build up molecular orbitals for multielectron molecules by forming Slater determinants of $H_2^+$-like molecular orbitals. The electronic Schrödinger
equation for $H_2^+$ is
$$
\hat H \psi_j(r_A, r_B, R) = E_j\psi_j(r_A, r_B,R)
$$
where the Hamiltonian operator (in atomic units) in the Born-Oppenheimer approximation is
$$
\hat H = -\frac 12 \nabla^2 - \frac 1{r_A} - \frac 1{r_B} + \frac 1R
$$
where $r_A$ and $r_B$ are the distances of the electron from nucleuss A and B, respectively, and R is the internuclear separation, which we treat as a fixed parameter.
We shall see that the molecular wave function spreads over both nuclei, and so we refer to it as a molecular orbital.
Figure 10.2 shows sketches of $\sigma$ states and $\pi$ states of $H_2^+$.
![QC-fig10.2]
The $\sigma$ state wave function is cylindrically symmetric, or has a circular cross section viewed along the internuclear axis. The $\pi$ state wave function has a cross section similar to an atomic p orbital.
Note that $\sigma$ orbitals have no nodal planes and that $\pi$ orbitals have one nodal plane containing the internuclear axis. It turns out that $\delta$ orbitals have two nodal planes containing the internuclear axis, and so on.
The states can be labeled by their behavior under an inversion of the molecule through its center of symmetry—that is, reflecting the positions of all the particles through the point midway between the nuclei.
Letting the inversion operator, called it $\hat P_{inv}$, operate on $\psi(r_A, r_B, R)$ gives
$$
\hat P_{inv} \psi(r_A, r_B, R) = c \psi(r_A, r_B, R)
$$
where $c= \pm 1$. Thus, the wave functions of $H_2^+$ either do not change or change in sign under inversion.
If the wave function remains unchanged, then the state is designated by a $g$ (for gerade, the German word for “even”); If it changes in sign, then the state is designated by a $u$ (for ungerade, the German word for “odd”).
![QC-fig10.3]
The $1\sigma_g$ means that this state is the first $\sigma_g$ state, in order of increasing energy around the equilibrium bond length for the ground state; the $\sigma_g1s$ means that this state dissociates to a proton and a hydrogen atom in a $1s$ state.
10.3 Molecular Orbitals Are Constructed from a Linear Combination of Atomic Orbitals
The method we will use to describe the bonding properties of molecules is called molecular orbital theory. In this case, the Schrödinger equation for the one-electron molecular ion $H_2^+$ is solved approximately using a linear combination of atomic orbitals, and the resulting orbitals are then used to construct determinantal wave functions for more complicated molecules.
It provides good physical insight into the nature of chemical bonds in molecules and yields results in good agreement with experimental observations.
Recall that the variational principle (Chapter 8) says that we can get a good approximation to the energy if we use an appropriate trial function. As a trial function for $\psi_j(r_A, r_B, R)$, we take the linear combination
$$
\psi = c_A 1s_A + c_B1s_B
$$
where $1s_A$ and $1s_B$ are hydrogen atomic orbitals centered on nuclei A and B, respectively. The molecular orbital given by this way is a linear combination of atomic orbitals, and is called an LCAO molecular orbital.
The $1s_A$ and $1s_B$ orbitals, from which the molecular orbital is constructed, are called basis functions, and constitute a basis set.
The secular equation is
$$
\left| \begin{matrix} H_{AA}-E &H_{AB}-ES \ H_{BA-ES} &H_{BB}-E \end{matrix} \right| =0
$$