Møller-Plesset perturbation theory - AbsoluteAstronomy.com

Møller–Plesset perturbation theory (MP) is one of several quantum chemistry

Quantum chemistry

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

post-Hartree–Fock ab initio methods

Ab initio quantum chemistry methods

Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...

in the field of computational chemistry

Computational chemistry

Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...

. It improves on the Hartree–Fock method by adding electron correlation

Electronic correlation

Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...

effects by means of Rayleigh–Schrödinger perturbation theory

Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

(RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order.
Its main idea was published as early as 1934 by Christian Møller

Christian Møller

Christian Møller was a Danish chemist and physicist who made fundamental contributions to the theory of relativity, theory of gravitation and quantum chemistry...

and Milton S. Plesset

Milton S. Plesset

Milton Spinoza Plesset was an American applied physicist who worked in the field of fluid mechanics and nuclear energy. He was elected to the National Academy of Engineering in 1979 for his fundamental contributions to multiphase flows, bubble dynamics, and safety of nuclear reactors...

Rayleigh–Schrödinger perturbation theory

The MP-theory is a special application of RS-PT. In RS-PT one considers an unperturbed Hamiltonian operator

to which is added a small (often external) perturbation

where λ is an arbitrary real parameter. In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The perturbation is the correlation potential.

In RS-PT the perturbed wave function and perturbed energy are expressed as a power series in λ:

Substitution of these series into the time-independent Schrödinger equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

gives a new equation: (

)

Equating the factors of

in this equation gives an kth-order perturbation equation, where k = 0, 1, 2, ..., n. See perturbation theory

Perturbation theory (quantum mechanics)

for more details.

Original formulation

The MP-energy corrections are obtained from Rayleigh–Schrödinger (RS) perturbation theory with the perturbation
(correlation potential):

where the normalized Slater determinant

Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...

Φ₀ is the lowest eigenfunction of the
Fock operator

Here N is the number of electrons of the molecule under consideration, H is the usual electronic Hamiltonian,

is the one-electron Fock operator, and ε_i is the orbital energy belonging to the doubly occupied spatial orbital φ_i.
The shifted Fock operator

serves as the unperturbed (zeroth-order) operator.

The Slater determinant Φ₀ being an eigenfunction of F, it follows
readily that

so that the zeroth-order energy is the expectation value of H with respect to Φ₀, i.e., the Hartree–Fock energy:

Since the first-order MP energy

is obviously zero, the lowest-order MP correlation energy appears in second order. This result is the Møller–Plesset theorem: the correlation potential does not contribute in first-order to the exact electronic energy.

In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-PT formula is written on basis of doubly excited Slater determinants. (Singly excited Slater determinants do not contribute because of the Brillouin theorem).
After application of the Slater–Condon rules for the simplification of N-electron matrix elements with Slater determinants in bra and ket and integrating out spin, it becomes

where φ_i and φ_j are canonical occupied orbitals and
φ_a and φ_b are canonical virtual orbitals. The quantities ε_i, ε_j, ε_a, and ε_b are the corresponding orbital energies. Clearly, through second-order in the correlation potential, the total electronic energy is given by the Hartree–Fock energy plus second-order MP correction: E ≈ E_HF + E_MP2. The solution of the zeroth-order MP equation (which by definition is the Hartree–Fock equation) gives the Hartree–Fock energy. The first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy.

Alternative formulation

Equivalent expressions are obtained by a slightly different partitioning of the Hamiltonian, which results in a different division of energy terms over zeroth- and first-order contributions, while for second- and higher-order energy corrections the two partitionings give identical results. The formulation is commonly used by chemists, who are now large users of these methods. This difference is due to the fact, well-known in Hartree–Fock theory, that

(The Hartree–Fock energy is not equal to the sum of occupied-orbital energies). In the alternative partitioning, one defines

Clearly in this partitioning,

Obviously, the Møller–Plesset theorem does not hold in the sense that E_MP1 ≠ 0. The solution of the zeroth-order MP equation is the sum of orbital energies. The zeroth plus first-order correction yields the Hartree–Fock energy. As with the original formulation, the first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy. We reiterate that the second- and higher-order corrections are the same in both formulations.

Use of Møller–Plesset perturbation methods

Second (MP2), third (MP3), and fourth (MP4) order Møller–Plesset calculations are standard levels used in calculating small systems and are implemented in many computational chemistry codes. Higher level MP calculations, generally only MP5, are possible in some codes. However, they are rarely used because of their cost.

Systematic studies of MP perturbation theory have shown that it is not necessarily a convergent theory at high orders. Convergence can be slow, rapid, oscillatory, regular, highly erratic or simply non-existent, depending on the precise chemical system or basis set.
The density matrix for the first-order and higher MP2 wavefunction is of the
type known as response density, which differs from the
more usual expectation value density. The eigenvalues of
the response density matrix (which are the occupation
numbers of the MP2 natural orbitals) can therefore be
greater than 2 or negative. Unphysical numbers are a sign of
a divergent perturbation expansion.

Additionally, various important molecular properties calculated at MP3 and MP4 level are no better than their MP2 counterparts, even for small molecules.

For open shell molecules, MPn-theory can directly be applied only to unrestricted Hartree–Fock reference functions (since ROHF states are not in general eigenvectors of the Fock operator). However, the resulting energies often suffer from severe spin contamination

Spin contamination

In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial parts of α and β spin-orbitals are permitted to differ....

, leading to large errors. A possible better alternative is to use one of the MP2-like methods based on restricted open-shell Hartree–Fock (ROHF). Unfortunately, there are many ROHF based MP2-like methods because
of arbitrariness in the ROHF wavefunction(for example HCPT, ROMP, RMP (also called ROHF-MBPT2), OPT1 and OPT2, ZAPT, IOPT, etc.). Some of the ROHF based MP2-like theories suffer from spin-contamination in their perturbed density and energies beyond second-order.

These methods, Hartree–Fock, unrestricted Hartree–Fock and restricted Hartree–Fock use a single determinant wave function. Multi-configurational self-consistent field

Multi-configurational self-consistent field

Multi-configurational self-consistent field is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate...

(MCSCF) methods use several determinants and can be used for the unperturbed operator, although not uniquely, so many methods, such as complete active space perturbation theory (CASPT2), and Multi-Configuration Quasi-Degenerate Perturbation
Theory (MCQDPT), have been developed. Unfortunately, MCSCF based methods are not without perturbation series divergences.

Rayleigh–Schrödinger perturbation theory

Original formulation

Alternative formulation

Use of Møller–Plesset perturbation methods

See also