Møller–Plesset perturbation theory (MP) is one of several
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
post–Hartree–Fock ab initio methods in the field of
computational chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
. It improves on the
Hartree–Fock method by adding
electron correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
Ato ...
effects by means of Rayleigh–Schrödinger
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
(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 (22 December 1904 in Hundslev, Als (island), Als14 January 1980 in Ordrup) was a Danish people, Danish chemist and physicist who made fundamental contributions to the theory of relativity, theory of gravitation and quantum chemi ...
and
Milton S. Plesset.
Rayleigh–Schrödinger perturbation theory
The MP perturbation theory is a special case of RS perturbation theory. In RS theory one considers an unperturbed
Hamiltonian operator
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, to which a small (often external) perturbation
is added:
:
Here, ''λ'' is an arbitrary real parameter that controls the size of the perturbation. In MP theory the zeroth-order
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
is an exact
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
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 mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
in λ:
:
:
Substitution of these series into the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
gives a new equation as
:
:
Equating the factors of
in this equation gives a ''k''th-order perturbation equation, where . See
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
for more details.
Møller–Plesset perturbation
Original formulation
The MP-energy corrections are obtained from Rayleigh–Schrödinger (RS) perturbation theory with the unperturbed Hamiltonian defined as the ''shifted''
Fock operator,
:
and the perturbation defined as the ''correlation potential'',
:
where the normalized
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
Φ
0 is the lowest eigenstate of the Fock operator:
:
Here ''N'' is the number of electrons in the molecule under consideration (a factor of 2 in the energy arises from the fact that each orbital is occupied by a pair of electrons with opposite spin),
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''.
Since the Slater determinant Φ
0 is an eigenstate of
, it follows readily that
:
i.e. the zeroth-order energy is the expectation value of
with respect to Φ
0, the Hartree-Fock energy. Similarly, it can be seen that ''in this formulation'' the MP1 energy
:
.
Hence, the first meaningful correction appears at MP2 energy.
In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-PT formula is written in a 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 Within computational chemistry, the Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the origina ...
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 virtual (or unoccupied) 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: . 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, with this alternative formulation, the Møller–Plesset theorem does not hold in the literal 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. To reiterate, 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 Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital ...
reference functions (since UHF states are not in general eigenvectors of the Fock operator). However, the resulting energies often suffer from severe
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 p ...
, leading to large errors. A possible better alternative is to use one of the MP2-like methods based on
restricted open-shell Hartree–Fock Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. It uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons. This is the sim ...
(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 (MCSCF) 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 (e.g., for mole ...
(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
Complete active space perturbation theory (CASPTn) is a multireference electron correlation method for computational investigation of molecular systems, especially for those with heavy atoms such as transition metals, lanthanides, and actinides. I ...
(CASPT2), and Multi-Configuration Quasi-Degenerate Perturbation
Theory (MCQDPT), have been developed. Unfortunately, MCSCF based methods are not without perturbation series divergences.
See also
*
Electron correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
Ato ...
*
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 whic ...
*
Post-Hartree–Fock
In computational chemistry, post-Hartree–Fock methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the repu ...
*
List of quantum chemistry and solid state physics software
Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock (HF) and some post-Hartree–Fock methods. They may also include density functional theory (DFT ...
References
Further reading
*
*
*
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*
{{DEFAULTSORT:Moller-Plesset Perturbation Theory
Computational chemistry
Post-Hartree–Fock methods
Theoretical chemistry