11 Introduction to Molecules

11.1 The Molecular Hamiltonian

For a molecule, we can decompose the Hamiltonian operator as:

\[\begin{equation} \hat{H} = \hat{K}_N +\hat{K}_{e} + \hat{V}_{NN} + \hat{V}_{eN} + \hat{V}_{ee} \tag{11.1} \end{equation}\]

where we have decomposed the kinetic energy operator into nuclear and electronic terms, \(\hat{K}_N\) and \(\hat{K}_e\), as well as the potential energy operator into terms representing the interactions between nuclei, \(\hat{V}_{NN}\), between electrons, \(\hat{V}_{ee}\), and between electrons and nuclei, \(\hat{V}_{eN}\). Each term can then be calculated using:

\[\begin{equation} \begin{aligned} \hat{K}_{N} &=-\sum_{i}^{\text{nuclei}}{\frac {\hbar ^{2}}{2M_{i}}}\nabla_{{{\mathbf {R}}_{i}}}^{2} \\ \hat {K}_{e} &=-\sum_{i}^{\text{electrons}}{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{{{\mathbf{r}}_{i}}}^{2} \\ \hat{V}_{{NN}} &= \sum _{i}\sum _{{j>i}}{\frac {Z_{i}Z_{j}e^{2}}{4\pi \varepsilon _{0}\left|{\mathbf {R}}_{i}-{\mathbf {R}}_{j}\right|}} \\ \hat {V}_{{eN}} &=-\sum _{i}\sum _{j}{\frac {Z_{i}e^{2}}{4\pi \varepsilon_{0}\left|{\mathbf {R}}_{i}-{\mathbf {r}}_{j}\right|}} \\ \hat{V}_{{ee}} &= \sum _{i}\sum _{{i<j}}{\frac {e^{2}}{4\pi \varepsilon _{0}\left|{\mathbf {r}}_{i}-{\mathbf {r}}_{j}\right|}}, \end{aligned} \tag{11.2} \end{equation}\]

where \(M_i\), \(Z_i\), and \(\mathbf{R}_i\) are the mass, atomic number, and coordinates of nucleus \(i\), respectively, and all other symbols are the same as those used in eq. (10.1) for the many-electron atom Hamiltonian.

11.1.1 Small terms in the molecular Hamiltonian

The operator in eq. (11.1) is known as the “exact” nonrelativistic Hamiltonian in field-free space. However, it is important to remember that it neglects at least two effects. Firstly, although the speed of an electron in a hydrogen atom is less than 1% of the speed of light, relativistic mass corrections can become appreciable for the inner electrons of heavier atoms. Secondly, we have neglected the spin-orbit effects, which is explained as follows. From the point of view of an electron, it is being orbited by a nucleus which produces a magnetic field (proportional to \({\bf L}\)); this field interacts with the electron’s magnetic moment (proportional to \({\bf S}\)), giving rise to a spin-orbit interaction (proportional to \({\bf L} \cdot {\bf S}\) for a diatomic.) Although spin-orbit effects can be important, they are generally neglected in quantum chemical calculations, and we will neglect them in the remainder of this textbook as well.

11.2 The Born-Oppenheimer Approximation

As we already saw in the previous chapter, if a Hamiltonian is separable into two or more terms, then the total eigenfunctions are products of the individual eigenfunctions of the separated Hamiltonian terms. The total eigenvalues are then sums of individual eigenvalues of the separated Hamiltonian terms.

For example. let’s consider a Hamiltonian that is separable into two terms, one involving coordinate \(q_1\) and the other involving coordinate \(q_2\):

\[\begin{equation} \hat{H} = \hat{H}_1(q_1) + \hat{H}_2(q_2) \tag{11.3} \end{equation}\]

with the overall Schrödinger equation being:

\[\begin{equation} \hat{H} \psi(q_1, q_2) = E \psi(q_1, q_2). \tag{11.4} \end{equation}\]

If we assume that the total wave function can be written in the form:

\[\begin{equation} \psi(q_1, q_2) = \psi_1(q_1) \psi_2(q_2), \tag{11.5} \end{equation}\]

where \(\psi_1(q_1)\) and \(\psi_2(q_2)\) are eigenfunctions of \(\hat{H}_1\) and \(\hat{H}_2\) with eigenvalues \(E_1\) and \(E_2\), then:

\[\begin{equation} \begin{aligned} \displaystyle \hat{H} \psi(q_1, q_2) &= ( \hat{H}_1 + \hat{H}_2 ) \psi_1(q_1) \psi_2(q_2) \\ &= \hat{H}_1 \psi_1(q_1) \psi_2(q_2) + \hat{H}_2 \psi_1(q_1) \psi_2(q_2) \\ &= E_1 \psi_1(q_1) \psi_2(q_2) + E_2 \psi_1(q_1) \psi_2(q_2) \\ &= (E_1 + E_2) \psi_1(q_1) \psi_2(q_2) \\ &= E \psi(q_1, q_2) \end{aligned} \tag{11.6} \end{equation}\]

Thus the eigenfunctions of \(\hat{H}\) are products of the eigenfunctions of \(\hat{H}_1\) and \(\hat{H}_2\), and the eigenvalues are the sums of eigenvalues of \(\hat{H}_1\) and \(\hat{H}_2\).

If we examine the nonrelativistic Hamiltonian in eq. (11.1), we see that the \(\hat{V}_{eN}\) terms prevents us from cleanly separating the electronic and nuclear coordinates and writing the total wave function. If we neglect these terms, we can write the total wave function as:

\[\begin{equation} \psi({\bf r}, {\bf R}) = \psi_e({\bf r}) \psi_N({\bf R}), \tag{11.7} \end{equation}\]

This approximation is called the Born-Oppenheimer approximation, and allows us to treat the nuclei as nearly fixed with respect to electron motion. The Born-Oppenheimer approximation is almost always quantitatively correct, since the nuclei are much heavier than the electrons and the (fast) motion of the latter does not affect the (slow) motion of the former. Using this approximation, we can fix the nuclear configuration at some value, \({\bf R_a}\), and solve for the electronic portion of the wave function, which is dependent only parametrically on \({\bf R}\) (we write this wave function as \(\psi_e \left({\bf r}; {\bf R_a} \right)\), where the semicolon indicate the parametric dependence on the nuclear configuration). To solve the TISEq we can then write the electronic Hamiltonian as:

\[\begin{equation} \hat{H}_{\text{e}} = \hat{K}_e({\bf r}) + \hat{V}_{eN}\left({\bf r}; {\bf R_a} \right) + \hat{V}_{ee}({\bf r}) \tag{11.8} \end{equation}\]

where we have also factored out the nuclear kinetic energy, \(\hat{K}_N\) (since it is smaller than \(\hat{K}_e\) by a factor of \(\frac{M_i}{m_e}\)), as well as \(\hat{V}_{NN}({\bf R})\). This latter approximation is justified, since in the Born-Oppenheimer approximation \({\bf R}\) is just a parameter, and \(\hat{V}_{NN}({\bf R_a})\) is a constant that shifts the eigenvalues only by some fixed amount. This electronic Hamiltonian results in the following TISEq:

\[\begin{equation} \hat{H}_{e} \psi_e \left({\bf r}; {\bf R_a} \right) = E_{e} \psi_e \left({\bf r}; {\bf R_a} \right), \tag{11.9} \end{equation}\]

which is the equation that is used to explain the chemical bond in the next section. Notice that eq. (11.9) is not the total TISEq of the system, since the nuclear eigenfunction and its eigenvalues (which can be obtained solving the Schrödinger equation with the nuclear Hamiltonian) are neglected. As a final note, in the remainder of this textbook we will confuse the term “total energy” with “total energy at fixed geometry”, as is customary in many other quantum chemistry textbooks (i.e., we are neglecting the nuclear kinetic energy). This is just \(E_{e}\) of eq. (11.9), plus the constant shift,\(\hat{V}_{NN}({\bf R_a})\), given by the nuclear-nuclear repulsion.

11.3 Solving the Electronic Eigenvalue Problem

Once we have invoked the Born-Oppenheimer approximation, we can attempt to solve the electronic TISEq in eq. (11.9). However, for molecules with more than one electron, we need to—once again—keep in mind the antisymmetry of the wave function. This obviously means that we need to write the electronic wave function as a Slater determinant (i.e., all molecules but \(\mathrm{H}_2^+\) and a few related highly exotic ions). Once this is done, we can work on approximating the Hamiltonian, a task that is necessary because the presence of the electron-electron repulsion term forbids its analytic treatment. Similarly to the many-electron atom case, the simplest approximation to solve the molecular electronic TISEq is to use the variational method and to neglect the electron-electron repulsion. As we noticed in the previous chapter, this approximation is called the Hartree-Fock method.

11.3.1 The Hartree-Fock Method

The main difference when we apply the variational principle to a molecular Slater determinant is that we need to build orbitals (one-electron wave functions) that encompass the entire molecule. This can be done by assuming that the atomic contributions to the molecular orbitals will closely resemble the orbitals that we obtained for the hydrogen atom. The total molecular orbital can then be built by linearly combine these atomic contributions. This method is called linear combination of atomic orbitals (LCAO). A consequence of the LCAO method is that the atomic orbitals on two different atomic centers are not necessarily orthogonal, and eq. (10.12) cannot be simplified easily. If we replace each atomic orbital \(\psi(\mathbf{r})\) with a linear combination of suitable basis functions \(f_i(\mathbf{r})\):

\[\begin{equation} \psi(\mathbf{r}) = \sum_i^m c_{i} f_i(\mathbf{r}), \tag{11.10} \end{equation}\]

we can then use the following notation:

\[\begin{equation} \displaystyle H_{ij} = \int \phi_i^* {\hat H} \phi_j d\mathbf{\tau}\;, \qquad \displaystyle S_{ij} = \int \phi_i^* \phi_jd\mathbf{\tau}, \tag{11.11} \end{equation}\]

to simplify eq. (10.12) to:

\[\begin{equation} E[\Phi] = \frac{\sum_{ij} c_i^* c_j H_{ij}}{\sum_{ij} c_i^* c_j S_{ij}}. \tag{11.12} \end{equation}\]

Differentiating this energy with respect to the expansion coefficients \(c_i\) yields a non-trivial solution only if the following “secular determinant” equals zero:

\[\begin{equation} \begin{vmatrix} H_{11}-ES_{11} & H_{12}-ES_{12} & \cdots & H_{1m}-ES_{1m}\\\ H_{21}-ES_{21} & H_{22}-ES_{22} & \cdots & H_{2m}-ES_{2m}\\\ \vdots & \vdots & \ddots & \vdots\\\ H_{m1}-ES_{m1} & H_{m2}-ES_{m2} & \cdots & H_{mm}-ES_{mm} \end{vmatrix}=0 \tag{11.13} \end{equation}\]

where \(m\) is the number of basis functions used to expand the atomic orbitals. Solving this set of equations with a Hamiltonian where the electron-electron correlation is neglected results is non-trivial, but possible. The reason for the complications comes from the fact that even if we are neglecting the direct interaction between electrons, each of them interact with the nuclei through an interaction that is screened by the average field of all other electrons, similarly to what we saw for the helium atom. This means that the Hamiltonian itself and the value of the coefficients \(c_i\) in the wave function mutually depend on each other. A solution to this problem can be achieved numerically using specialized computer programs that use a cycle called the self-consistent-field (SCF) procedure. Starting from an initial guess of the coefficients, an approximated Hamiltonian operator is built from them and used to solve eq. (11.13). This solution gives updated values of the coefficients, which can then be used to create an improved version of the approximated Hamiltonian. This procedure is repeated until both the coefficients and the operator do not change anymore. From this final solution, the energy of the molecule is then calculated.