NationMaster - Encyclopedia: Molecular Hamiltonian

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. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ... The electron is a fundamental subatomic particle that carries an electric charge. ... A semi-accurate depiction of the helium atom. ... In chemistry a polyatomic molecule is a molecule that consists only of atoms of a single element. ...

In contrast with the molecular Hamiltonian, the electronic Hamiltonian does not include the kinetic energy operator corresponding to the contributions from the nuclei. Kinetic energy is the energy that a body possesses as a result of its motion. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...

1 Mathematical expression
2 Adiabatic formalism or Born-Oppenheimer approximation
3 See also
4 References

Mathematical expression

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

The kinetic energy operators for each nucleus in the system;
The kinetic energy operators for each electron in the system;
The potential energy between the electrons and nuclei - the total electron-nucleus Coulombic attraction in the system;
The potential energy arising from Coulombic electron-electron repulsions
The potential energy arising from Coulombic nuclei-nuclei repulsions - also known as the nuclear repulsion energy. See electric potential for more details.

$hat{T}_n = - sum_i frac{hbar^2}{2 M_i} frac{partial^2}{partial R_i^2}$
$hat{T}_e = - sum_i frac{hbar^2}{2 m_e} frac{partial^2}{partial r_i^2}$
$hat{U}_{en} = - sum_i sum_j frac{Z_i e^2}{4 pi epsilon_0 left | R_i - r_j right | }$
$hat{U}_{ee} = {1 over 2} sum_i sum_{j ne i} frac{e^2}{4 pi epsilon_0 left | r_i - r_j right | }$
$hat{U}_{nn} = {1 over 2} sum_i sum_{j ne i} frac{Z_i Z_j e^2}{4 pi epsilon_0 left | R_i - R_j right | }$

Where R_i and r_i are the position vectors of the i'th nucleus and electron. M_i is the mass of the nucleus and m_e is the mass of the electron Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ...

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 $mathbf{R}$ . A potential energy surface is generally used within the adiabatic or Bornâ€“Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems. ...

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

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

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 In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

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

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.

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: Max Born Max Born (December 11, 1882 in Breslau - January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... J. Robert Oppenheimer, the father of the atomic bomb served as the first director of Los Alamos National Laboratory, beginning in 1943. ...

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. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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:

Assume that the nuclei only make small oscillations from equilibrium configuration so the vibrational potential can be considered harmonic;
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." The rigid rotor is a mechanical model that is used to explain rotating systems. ...

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 H_M 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 Look up Portmanteau word in Wiktionary, the free dictionary. ...

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

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 $(x e, X n)$ 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 $3 N - 5$ or $3 N - 6$ vibrations if $N$ is the number of nuclei, and whether the molecule is linear or nonlinear.

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 $E a$ refers to the energy of the state $ψ a (x e, X n)$ . 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 $ψ a (x e, X n)$ to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

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

where $e,ν$ is the electronic and nuclear vibration quantum number. This formulation is termed an adiabatic wavefunction. A quantum number describes the energies of electrons in atoms. ...

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 $X n = {X 1, X 2,... X 3 N - 6}$ or $X n = {X 1, X 2,... X 3 N - 5}$ 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 $X n$ . The electronic wavefunctions defines a complete ortonomal set of functions for each $X n$ so the molecular wavefunction can be expanded in the basis.

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

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 cooeficients $χ e,ν (X n)$ describes the vibrational eigenfunctions and the $E e (X n)$ describe the vibrational potential energy. The eigenvalue, $E e (X n)$ is often approximated by an harmonic function for simplification.

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 electron is a fundamental subatomic particle that carries an electric charge. ... Normals modes of vibration progression through a crystal. ... In theoretical chemistry, the vibronic coupling terms (which are neglected within the Born-Oppenheimer approximation) are proportional to the interaction between electronic and nuclear motions of molecules. ... The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold of N-1 dimensions. ... In quantum chemistry, a conical intersection of two potential energy surfaces of the same spatial and spin symmetries is the set of molecular geometry points where the two potential energy surfaces are degenerate (intersect). ...

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 $T n$ is the nuclear kinetic energy operator and the electronic wavefunction $φ e$ is parametrically (not explicitly) dependent on the nuclear coordinates.

References

Born, Max, Oppenheimer,Robert (25 August 1927). "Zur Quantentheorie der Molekeln". Annalen der Physik 389: 457-484.
Moss, R. E. (1973). Advanced Molecular Quantum Mechanics. Chapman and Hall.
Tinkham, Michael (2003). Group Theory and Quantum Mechanics. Dover Publications. ISBN 0-146-43247-5.
C. Handy, Nicholas, Yamaguchi, Yukio,F. Schaefer, III, Henry (1986). "The diagonal correction to the Born-Oppenheimer approximation: Its effect on the singlet-triplet splitting of CH₂ and other molecular effects". J. Chem. Phys. 84: 4481.

Categories: Molecular physics | Quantum chemistry Max Born Max Born (December 11, 1882 in Breslau - January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... J. Robert Oppenheimer, the father of the atomic bomb served as the first director of Los Alamos National Laboratory, beginning in 1943. ... Chapman and Hall was a British publishing house, founded in the first half of the 19th century by Edward Chapman and William Hall. ... Dover Publications is a book publisher founded in 1941. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Henry F. Schaefer III was born in Grand Rapids, Michigan in 1944. ...

Results from FactBites:

Molecular Hamiltonian - Wikipedia, the free encyclopedia (1050 words)
	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.
	Most treatments of the Electronic molecular Hamiltonian use the adiabatic formalism originally discussed by Born and Oppenheimer.

Electronic Hamiltonian - Encyclopedia, History, Geography and Biography (639 words)
	The Electronic Hamiltonian is an operator in quantum mechanics (and in particular quantum chemistry) which describes the motions of electrons and nuclei in a polyatomic molecule.
	The electronic molecular Hamiltonian is the term of the molecular Hamiltonian obtained when the molecular geometry is frozen.
	Within the Born-Oppenheimer approximation, the electronic Hamiltonian is said to depend adiabatically on the molecular geometry.

More results at FactBites »

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