Empirical Parameterization of Diatomic Molecules

The main features of chemical bonding by electron pairs are properly described by the HL model of $H_2$ (see page 91). According to such model, the covalent bond is described by a singlet state,

$$^1\psi_{HL} = N_1[\alpha(1)\beta(2)- \beta(1) \alpha(2)][\chi_A(1)\chi_B(2) +\chi_A(2)\chi_B(1)],$$

with energy

$$^1E_+ = <^1\psi_{HL}\vert H\vert ^1\psi_{HL}> =\frac{J+K}{1+S^2},$$

where $H=h(1) + h(2) +e^2/r_{12},$ with

$h(1)= -\frac{\hbar^2}{2m} \nabla_1^2 -\frac{e^2}{r_{1A}}-\frac{e^2}{r_{1B}},$

$h(2)= -\frac{\hbar^2}{2m} \nabla_2^2 -\frac{e^2}{r_{2A}}-\frac{e^2}{r_{2B}},$

$J=<\chi_A(1)\chi_B(2)\vert H\vert\chi_A(1)\chi_B(2)> \qquad$ Coulomb integral

$K=<\chi_A(1)\chi_B(2)\vert H\vert\chi_A(2)\chi_B(1)> \qquad$ Exchange integral

$S^2=<\chi_A(1)\chi_B(2)\vert\chi_A(2)\chi_B(1)>.$

Similarly, the triplet state is described as follows,

$$^3\psi_{HL}= N_3[\chi_A(1)\chi_B(2) -\chi_B(1)\chi_A(2)] \begin{c... ...2)+ \beta(1) \alpha(2)]\\ \alpha(1) \alpha(2) \\ \beta(1) \beta(2) \end{cases},$$

and has energy

$$^3E_-= \frac{(J-K)}{(1-S^2)}.$$

The energies of the singlet and triplet states are parametrized by the internuclear H-H distance and can be represented by the following diagram,

The energies $^1E$ and $^3E$ can be approximated by the following analytical functions:

$$^1E_+ \approx D\left[ e^{-2a(R-R_0)}-2 e^{-a(R-R_0)} \right] \equiv M(R),$$

$$^3E_- \approx \frac{D}{2}\left[ e^{-2a(R-R_0)} +2 e^{-a(R-R_0)} \right] \equiv M^*(R).$$

Parameters $D$ and $a$ can be obtained by fitting M(R) to the actual (experimental or ab-initio) ground state potential energy surface. Such parametrization allows us to express the Coulombic and Resonance integrals J and K in terms of available experimental (or ab initio) data as follows,

$$J\approx \frac{1}{2} [ (M+M^*) +S^2(M-M^*) ],$$

$$K\approx \frac{1}{2} [ (M-M^*) +S^2(M+M^*) ].$$

This parametrization of Hamiltonian matrix elements illustrates another example of semi-empirical parametrization that can be implemented by using readily available experimental information (remember that in the previous section we described the semiempirical parametrization of the Hückel model according to the absorption spectrum of the molecule).

The covalent nature of the chemical bond significantly changes when one of the two atoms in the molecule is substituted by an atom of different electronegativity. Under those circumstances, the wave function should include ionic terms, e.g.,

$^1\psi_A^{ion} = \tilde{N} \chi_A(1) \chi_A(2) [\alpha(1)\beta(2)-\beta(1) \alpha(2)],$

and

$^1\psi_B^{ion} = \tilde{N} \chi_B(1) \chi_B(2) [\alpha(1)\beta(2)-\beta(1) \alpha(2)].$

The complete wave function (with both covalent and ionic terms) can be described as follows, $\psi=C_1 \psi_1 +C_2 \psi_2$, where the covalent wave function is

$\psi_1 = [\alpha(1)\beta(2)- \beta(1) \alpha(2)](\chi_A(1)\chi_B(2) +\chi_A(2)\chi_B),$

and the ionic wave function is

$\psi_2 = [\alpha(1)\beta(2)- \beta(1) \alpha(2)][\chi_A(1)\chi_A(2)\xi_1 +\chi_B(1)\chi_B(2)(1-\xi_1)],$

where the parameter $\xi_1$ is determined by the relative electronegativity of the two atoms. For example, consider the HF molecule. For such molecule $\xi_1$ =1, A represents the F atom, and B represents the H atom (i.e., due to the electronegativity difference between the two atoms, the predominant ionic configuration is $H^+F^-$). Therefore, the ground state energy $E_g$ is obtained as the lowest eigenvalue of the secular equation,

$$\left \vert \begin{matrix}H_{11}-E & H_{12} \\ H_{12} & H_{22}-E \\ \end{matrix} \right \vert =0.$$ (66)

Here we have neglected $S_{12}$, assuming that such approximation can be partially corrected according to the parametrization of $H_{12}$. The semiempirical parametrization strategy can be represented by the following diagram:

This diagram represents the following curves:

$H_{11}=\bar{M} = \bar{D} [e^{-2a(R-R_0-\delta)} -2 e^{-a(R-R_0-\delta)}]$ is a covalent state represented by a Morse potential $\bar{M}$.

$H_{22}=I -EA - \frac{332}{R} + A e^{-bR} +C R^{-9},$ is the potential energy surface of the ionic state, where the difference between the H ionization energy and the F electron affinity, I-EA, corresponds to the energy of forming the ion pair $H^+$ $F^-$. The term $-\frac{332}{R}$ is the Coulombic interaction and $A e^{-bR} +C R^{-9}$ is the short range repulsive potential.

The ground state potential energy surface $E_g=M=D[e^{-2a(R-R_0)} -2 e^{-a(R-R_0)}]$ is represented by a Morse potential $M$. Parameters $D$ and $R_0$ can be obtained from the experimental bond-energy and bond-length. The parameter $a$ can be adjusted to reproduce the vibrational frequency of the diatomic molecule. The parameter $\bar{D}_{HF} =\sqrt{D_{HH} D_{FF}}$   and$\delta = 0.05 \AA$. Parameters A and C are adjusted so that the minimum energy of $H_{22}$ corresponds to the H-F bond-length (i.e., the sum of ionic radii of H and F). This empirical parametrization allows us to solve Eq. (66) for $H_{12}$,

$$H_{12}=\sqrt{(H_{11}-M)(H_{22}-M)},$$

and obtain the Hamiltonian matrix elements in terms of empirical parameters.

Conclusion: Potential energy surfaces parametrized by a few empirical parameters are able to describe bonding properties of molecules associated with atoms of different electronegativity.

Dipole Moment

The dipole moment is one of the most important properties of molecules and can be computed as follows,

$$\mu_g= <\psi_g\vert\hat{\mu}\vert\psi_g>,$$

where

$$\hat{\mu}= -\sum_i e r_i + \sum_j ez_j R_j.$$

The first term of this equation involves electronic coordinates $r_i$ and the second term involves nuclear coordinates $R_j$.

For example, the dipole moment of $HF$ can be computed as follows,

$$\mu_g= C_1^2 \underbrace{<\psi_1\vert\hat{\mu}\vert\psi_1>}_{0} +... ...rt\psi_2>}_{eR_0} + 2C_1C_2 \underbrace{<\psi_1\vert\hat{\mu}\vert\psi_2>}_{0},$$

since $\psi_1$ represents a covalent state and the overlap between $\psi_1$ and $\psi_2$ is assumed to be negligible.

The dipole moment is usually reported in Debye units, where 4.803 Debye is the dipole moment of two charges of 1 a.u. with opposite sign and separated by 1 Å, from each other.

Exercise 56: Evaluate the dipole moment for HF using the following parameters for the semiempirical model of HF potential energy surfaces (energies are expressed in kcal/mol, and distances in Å),

D=134;         $\bar{D}$=61;         $R_0$=0.92;         a=2.27;

A=640;         b=2.5;         C=20;         I=313;         EA=83.

Polarization

The electric field of an external charge $z$ located at coordinate $R_0$ along the axis of the molecule does not affect the energy of the covalent state $H_{11}$, but affects the energy of the ionic state $H_{22}$ as follows,

$$H_{22}' = H_{22} + \frac{ze}{R_{H^+C}} - \frac{ze}{R_{F^-C}}.$$

Therefore, the presence of an external charge perturbs the ground state energy of the molecule. Such perturbation can be computed by re-diagonalizing Eq. (66), using $H_{22}'$ instead of $H_{22}$. Solving for the ground state energy we obtain,

$$E'_g = \frac{1}{2} \left [ (H'_{22}+H_{11}) - ((H'_{22}- H_{11})+4H_{12}^2)^{1/2} \right ].$$

Exercise 57:

(1) Plot $E_g$, as a function of the internuclear distance $R$, for the HF molecule in the presence of an external charge located in the axis of the molecule at 10 Å, to the left of the F atom.

(2) Compare your results with the analog Gaussian98 calculation by using the scan keyword. Hint: The Gaussian98 input file necessary to scan the ground state potential energy surface of $H_2$ is described as follows,

$$\begin{array}{ll} \hspace{-6.cm} \char93 \text{hf/6-31G scan... ...m} \vspace{-.20cm} \text{potential scan for H}_2 & \end{array}$$

$$\begin{array}{llll} \hspace{-6.cm} & & & \\ \hspace{-6.cm} 0... ...} & & &\\ \hspace{-6.cm} \text{R} & 0.9 & 5 & 0.1 \end{array}$$

This input file scans the potential energy of $H_2$ by performing single point calculations at 5 internuclear distances. The output energies are represented by the following diagram: