Molecular Hamiltonian

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The molecular Hamiltonian is an operator in quantum chemistry and atomic, molecular, and optical physics which describes the motions of electrons and nuclei in a polyatomic molecule.

In contrast with the molecular Hamiltonian, the electronic Hamiltonian does not include the kinetic energy operator corresponding to the contributions from the nuclei.

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[edit] Mathematical expression

The molecular Hamiltonian is a sum of 5 terms. They are

  1. The kinetic energy operators for each nucleus in the system;
  2. The kinetic energy operators for each electron in the system;
  3. The potential energy between the electrons and nuclei - the total electron-nucleus Coulombic attraction in the system;
  4. The potential energy arising from Coulombic electron-electron repulsions
  5. The potential energy arising from Coulombic nuclei-nuclei repulsions - also known as the nuclear repulsion energy. See electric potential for more details.
  1. \hat{T}_n = - \sum_i \frac{\hbar^2}{2 M_i} \frac{\partial^2}{\partial R_i^2}
  2. \hat{T}_e = - \sum_i \frac{\hbar^2}{2 m_e} \frac{\partial^2}{\partial r_i^2}
  3. \hat{U}_{en} = - \sum_i \sum_j \frac{Z_i e^2}{4 \pi \epsilon_0 \left | R_i - r_j \right | }
  4. \hat{U}_{ee} = {1 \over 2} \sum_i \sum_{j \ne i} \frac{e^2}{4 \pi \epsilon_0 \left | r_i - r_j \right | }
  5. \hat{U}_{nn} = {1 \over 2} \sum_i \sum_{j \ne i} \frac{Z_i Z_j e^2}{4 \pi \epsilon_0 \left | R_i - R_j \right | }

Where Ri and ri are the position vectors of the i'th nucleus and electron. Mi is the mass of the nucleus and me is the mass of the electron

The electronic Hamiltonian is defined to be

\hat{H}_e = \hat{T}_e + \hat{U}_{en} + \hat{U}_{nn} + \hat{U}_{ee}

so that the molecular Hamiltonian is written as

\hat{H} = \hat{H}_e + \hat{T}_n.

The electronic Hamiltonian contains all three potential terms because their sum

\hat{V} \left ( \mathbf{r} , \mathbf{R} \right ) = \hat{U}_{en} + \hat{U}_{nn} + \hat{U}_{ee}

is the expression which gives rise to the physically meaningful potential energy surfaces ubiquitous in chemistry, for fixed nuclear geometry \mathbf{R}.

In laboratory reference frame, the Hamiltonians are given by

\hat{H}_{M,lab}=\hat{T}_{e,lab}+\hat{T}_{n,lab}+V(\mathbf{r},\mathbf{R})

in which

\hat{T}_{e,lab}=\frac{-\hbar^2}{2 m_e}\sum_i \nabla^2_{i,lab}

refers to the energy of the electrons and

\hat{T}_{n,lab}=\frac{-\hbar^2}{2}\sum_a M_a^{-1}\nabla^2_{a,lab}

refers to the energy of the nuclei. The potential energy V(\mathbf{r},\mathbf{R}) describes all coulomb interactions between all pairs of charged molecules, which gives

V(\mathbf{r},\mathbf{R})=\frac{1}{4 \pi \epsilon_0}\left( \sum_{i<j}\frac{e^2}{\left| \mathbf{r}_i- \mathbf{r}_j\right|}+ \sum_{a<b}\frac{Z_a Z_b}{\left| \mathbf{R}_a - \mathbf{R}_b \right|} - \sum_{i,a}\frac{Z_a e^2}{\left| \mathbf{r}_i -\mathbf{R}_a\right|}\right)

This is the fundamental Hamiltonian describing an isolated molecule without spin.

If ones take the spin from electrons and nuclei into account, the molecular Hamiltonian for the isolated molecule is given by

\hat{H}=\hat{H}_M+\hat{H}_{nucl.spin}+\hat{H}_{elec.spin}

in which \hat{H}_M is the spinless molecular Hamiltonian describing the total translational energy contribution from all the nuclei and electrons. The last two terms describes the contributions from the spin of the nuclei and electrons.

[edit] Adiabatic formalism or Born-Oppenheimer approximation

Most treatments of the Electronic molecular Hamiltonian use the adiabatic formalism originally discussed by Born and Oppenheimer. The formalism applies an assumption based on the fact that nuclei are between 104 to 105 times larger than the electrons. Under these conditions, the nuclei on average move around much slower than the electron, and so the motions of electron and nuclei can be treated separately. This makes the spinless Schrödinger equation more tractable. The separation of variables require 3 steps:

  • Separation of the translational motion of the center of mass
  • Separation of the rotational motion
  • Separation of the electronic motion from the nuclear vibrations

The molecular Hamiltonian transformed into the rotating coordinate system with origin in the center of mass of the nuclei is given by

\hat{H}_M=\hat{T}_e + V(\mathbf{r},\mathbf{R}) +\hat{T}_n +\frac{1}{2} \hat{\mathbf{J_n}} I_n^{-1} \hat{\mathbf{J_n}} - \frac{ \hbar^2}{ 2 \sum_{i} M_i } \nabla^2_M+\frac{\left( -i \hbar \sum_i \frac{\partial}{\partial x_i}\right)^2}{2M_n}

The first term is the electronic kinetic energy, and the second term is the potential energy in the laboratory reference frame. The third term is the nuclear kinetic energy, and the fourth term is the rotational kinetic energy plus the coupling between rotations and vibrations. The fifth term represents the translational kinetic energy of the molecule, and the last term is the mass polarization term. For most practical purposes, it can be assumed to be small and is neglected. In the rotational term \frac{1}{2} \hat{\mathbf{J_n}} I_n^{-1} \hat{\mathbf{J_n}}, I_n^{-1} is the nuclear inertia tensor, and the \hat{\mathbf{J_n}} vector operator is related to the angular momentum of the vibrational motion.

[edit] Rotational Hamiltonian

Pure rotational spectra are very hard to achieve experimentally, but they can be described by further separation of the vibrational and electronic motions. This requires two things:

  1. Assume that the nuclei only make small oscillations from equilibrium configuration so the vibrational potential can be considered harmonic;
  2. Approximate the inertia tensor with the inertia tensor I_{n,eq} \,\; calculated at the equilibrium configuration.

This is also called the "Harmonic vibrational and rigid-rotor model."

[edit] Vibronic Hamiltonian

This is the most prevalent form of the molecular Hamiltonian because the vibrations are essentially independent of the surroundings. Hence, vibrational transitions are easily observed. Since the rotational transitions are almost never observed, a good approximation to the molecular Hamiltonian would be obtained by keeping only the part of HM that describes the electronic and vibrational parts. This is called the vibronic Hamiltonian, a portmanteau of "vibrational" and "electronic". The vibronic Hamiltonian is given by

\hat{H}_{M,vibronic}=\hat{T}_{e,vibronic}+\hat{T}_{n,vibronic}+V(x_e,X_n)

with

\hat{T}_{e,vibronic}=\frac{-\hbar^2}{2m_e}\sum_{x_e} \hat{\nabla}^2_{x_e}\quad \mathrm{and} \quad \hat{T}_{n,vibronic}=\frac{-\hbar^2}{2}\sum_{X_n} \frac{\hat{\nabla}^2_{X_n}}{M_{X_n}}

with the (xe,Xn) being internal electronic and nuclear vibration coordinates. The use of the internal coordinates is used since the coulomb interaction only depends on the relative distance between the charged particles. Since the rotational and translational motions are now separated there will be either 3N − 5 or 3N − 6 vibrations if N is the number of nuclei, and whether the molecule is linear or nonlinear.

[edit] Solving the molecular Schrödinger equation

The molecular Schrödinger equation is given by

\hat{H}_M \psi_a(x_e,X_n)=E_a \psi_a(x_e,X_n)

where Ea refers to the energy of the state ψa(xe,Xn). To solve the Schrödinger equation it is needed to decouple the motion of the nuclei and electrons. This is done by approximating the molecular wavefunction ψa(xe,Xn) to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by

\psi_a(x_e,X_n)=\phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)

where e,ν is the electronic and nuclear vibration quantum number. This formulation is termed an adiabatic wavefunction.

There are two main cases used in molecular physics, a dynamic and a static type. The dynamic type the electronic wavefunctions are assumed to follow the vibrations of the nuclei. The static case uses a static reference configuration to calculate the electronic wavefunctions, this is also called the crude adiabatic approximation.

In the dynamic approximation the electronic wavefunction is defined as the solution to the electronic Schrödinger equation

\hat{H}_e \phi_e(x_e,X_n)=E_e(X_n)\phi(x_e,X_n)

where

\hat{H}_e=\hat{H}_M-\hat{T}_n

with the electronic wavefunctions found the nuclear vibrational coordinates Xn = {X1,X2,...X3N − 6} or Xn = {X1,X2,...X3N − 5} can be treated as parameters and the solution of the electronic Schrödinger equation then define the dependence of the electronic wavefunction and eigenvalues on the set of nuclear vibration coordinates Xn. The electronic wavefunctions defines a complete orthonomal set of functions for each Xn so the molecular wavefunction can be expanded in the basis.

\psi_a(x_e,X_n)=\sum_{e,\nu} \phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)

using this result in the most used vibronic case, and inserting in the electronic Schrödinger equation and neglecting electronic coupling gives a new eigenvalue equation given by

(\hat{T}_{n,vibronic}+E_{e'}(X_n))\chi_{e',\nu'}(X_n)=E_{e',\nu'}\chi_{e',\nu'}(X_n)

where the expansion coefficients χe,ν(Xn) describes the vibrational eigenfunctions and the Ee(Xn) describe the vibrational potential energy. The eigenvalue, Ee(Xn) is often approximated by an harmonic function for simplification.

[edit] Limitations

When the assumptions required for the adiabatic Born-Oppenheimer approximation do not hold, the approximation is said to "break down". Other approaches are needed to properly describe the system which is beyond the Born-Oppenheimer approximation.

The explicit consideration of the coupling of electronic and nuclear (vibrational) movement is known as electron-phonon coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as vibronic coupling which is important in the case of avoided crossings or conical intersections.

The so-called 'diagonal Born-Oppenheimer correction' (DBOC) can be obtained as

\langle \phi_e(x_e,X_n)|T_n|\phi_e(x_e,X_n)\rangle

where Tn is the nuclear kinetic energy operator and the electronic wavefunction φe is parametrically (not explicitly) dependent on the nuclear coordinates.

[edit] See also

[edit] References

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