Electron Dynamics of Interfacial Electron Transfer

From WikidChem

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Propagation of an electron with the time-evolution operator

A wavepacket is propagated by applying the time-evolution operator,

<math>\Psi (t=\tau) = e^\frac{i\hat{H}\tau}{\hbar} \Psi (t=0)</math>,

starting from an initial wavepacket that is a linear combination of molecular orbitals for the system,

<math>\Psi (t=0) = \sum_{i=1}^n a_i \phi_i </math>,

where each molecular orbital has a corresponding energy,

<math>\hat{H}\phi_i = E_i\phi_i</math>.

Then the time evolved wavepacket is obtained by time-evolving the expansion coefficients,

<math>\Psi (t=\tau) = \sum_{i=1}^n e^\frac{iE_i\tau}{\hbar} a_i (t=0) \phi_i = \sum_{i=1}^n a_i (t=\tau) \phi_i </math>.

This shows that the we need to find the molecular orbitals and their energies for the wavepacket propagation. Other propagation methods are available that do not require knowledge of the molecular orbitals or energies.

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extended Hueckel method

In the EH method, the secular equation,

<math>H \phi = eS \phi</math>,

is solved to obtain the eigenvectors (molecular orbitals) <math>\phi</math> and eigenvalues (energies) e. H is the Hamiltonian of the system, which is approximated using adjustable parameters making the method semiemperical. The diagonal elements of the Hamiltonian matrix are approximated as,

<math>H_{ij} = IE_{i}</math>,

where IE is the ionization energy of atom i. The off-diagonal elements are approximated as,

<math>H_{ij} = \frac{1}{2} \beta (H_{ii} + H_{jj}) S_{ij}</math>

where <math>\beta</math> is an empirical parameter and <math>S_{ij}</math> is the overlap matrix element. Calculation of the overlap matrix is the most computationally challenging part of an extended Hueckel calculation.

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Calculation of overlap integrals in basis of Slater-type orbitals

A Slater-type orbital (STO) is of the form,

<math>\Psi_{n,l,m} = N r^{n-1}e^{-\xi r} Y_l^m (\theta,\phi)</math>,

where <math>N</math> is the normalization factor and <math>Y_l^m</math> is the spherical harmonic. The spherical harmonic can be expressed in terms of a Legendre function,

<math> Y_l^m (\theta,\phi) = N_a e^{im\phi} P_l^m (\cos\theta)</math>.

The overlap matrix element is of the form,

<math> S_{i,j} = \langle\Psi_i\vert\Psi_j\rangle = \int\limits_{V} N_i r_i^{n_i-1}e^{-\xi_i r_i} Y_{l_i}^{m_i} (\theta_i,\phi_i) N_j r_j^{n_j-1}e^{-\xi_j r_j} Y_{l_j}^{m_j} (\theta_j,\phi_j)\, dV</math>.

There are two types of overlap integrals; one-center, <math>r_i = r_j</math>, and two-center, <math>r_i\neq r_j</math>.