Distributed Multipole Analysis
Encyclopedia
Distributed multipole analysis (DMA) is a compact and accurate way of describing the spatial distribution of electric charge
within a molecule
.
The DMA method was devised by Prof. Anthony Stone of Cambridge University to describe the charge distribution of a molecule in terms of a multipole expansion
around a number of centers. The idea of using a multi-center multipole expansion was earlier proposed by Robert Rein. Typically, the centers correspond to the atoms constituting the molecule, though this is not a requirement. A multipole series, consisting of a charge
, dipole, quadrupole
and higher terms is located at each center. Importantly, the radius of convergence
of this multipole series is sufficiently small that the relevant series will be convergent when describing two molecules in van der Waals contact.
The DMA series are derived from ab initio
or density functional theory
calculations using Gaussian basis sets. If the molecular orbitals are written as linear combinations of atomic basis functions the electron density takes the form of a sum of products of the basis functions, called density matrix elements. Boys (1950) showed that the product of two spherical Gaussian functions, centered at different points, can be expressed as a single Gaussian at an intermediate point known as the overlap center.
If a basis
of Gaussian functions is used, the product of two s functions is spherically symmetric and can be represented completely just by a point charge at the ‘overlap center’ of the two Gaussian functions. The product of an s orbital and a p orbital has only charge and dipole components, and the product of two p functions has charge, dipole and quadrupole components.
If the overlap center is not at an atom, one can move the origin of the multipole expansion to the nearest distributed multipole site, re-expressing the series to account for the change of origin. The multipole expansion will no longer terminate, but the higher terms will be small. One may take the sites wherever one chooses, but they will usually be at the atoms. For small molecules one may wish to use additional sites at the centers of bonds; for larger molecules one may use a single site to describe a group of atoms such as a methyl group. The DMA procedure is exact and very fast, but for modern large basis sets with diffuse basis functions it has to be modified somewhat. When the basis functions have exponents that are small, the product function extends over several atoms, and it is better to calculate the distributed multipoles by numerical quadrature over a grid of points. The grid can be defined so that each point is associated with a particular site, and the multipoles for each site are obtained by quadrature over the points belonging to that site.
This description then includes at each site:
The DMA describes the potential at points outside the molecule
with an accuracy which is essentially that of the wavefunction
, so that its use entails no loss of precision. The DMA description gives the electrostatic energy of interaction between two molecules. It does not account for charge overlap effects and hence excludes the penetration energy.
DMA is inherently much more accurate than the commonly used partial charge
methodologies for calculating intermolecular interaction energies, since it captures anisotropy
of the atom-atom contributions to electrostatic interaction. It may therefore seem surprising that it has not been more widely used in molecular simulation. Possible reasons for this are:
DMA has found extensive use in crystal structure prediction
for small organic molecules
, where significant progress can often be made while using rigid molecular structures.
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
within a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
.
The DMA method was devised by Prof. Anthony Stone of Cambridge University to describe the charge distribution of a molecule in terms of a multipole expansion
Multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
around a number of centers. The idea of using a multi-center multipole expansion was earlier proposed by Robert Rein. Typically, the centers correspond to the atoms constituting the molecule, though this is not a requirement. A multipole series, consisting of a charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
, dipole, quadrupole
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
and higher terms is located at each center. Importantly, the radius of convergence
Radius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
of this multipole series is sufficiently small that the relevant series will be convergent when describing two molecules in van der Waals contact.
The DMA series are derived from ab initio
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...
or density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
calculations using Gaussian basis sets. If the molecular orbitals are written as linear combinations of atomic basis functions the electron density takes the form of a sum of products of the basis functions, called density matrix elements. Boys (1950) showed that the product of two spherical Gaussian functions, centered at different points, can be expressed as a single Gaussian at an intermediate point known as the overlap center.
If a basis
Basis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...
of Gaussian functions is used, the product of two s functions is spherically symmetric and can be represented completely just by a point charge at the ‘overlap center’ of the two Gaussian functions. The product of an s orbital and a p orbital has only charge and dipole components, and the product of two p functions has charge, dipole and quadrupole components.
If the overlap center is not at an atom, one can move the origin of the multipole expansion to the nearest distributed multipole site, re-expressing the series to account for the change of origin. The multipole expansion will no longer terminate, but the higher terms will be small. One may take the sites wherever one chooses, but they will usually be at the atoms. For small molecules one may wish to use additional sites at the centers of bonds; for larger molecules one may use a single site to describe a group of atoms such as a methyl group. The DMA procedure is exact and very fast, but for modern large basis sets with diffuse basis functions it has to be modified somewhat. When the basis functions have exponents that are small, the product function extends over several atoms, and it is better to calculate the distributed multipoles by numerical quadrature over a grid of points. The grid can be defined so that each point is associated with a particular site, and the multipoles for each site are obtained by quadrature over the points belonging to that site.
This description then includes at each site:
- Charges, describing electronegativityElectronegativityElectronegativity, symbol χ , is a chemical property that describes the tendency of an atom or a functional group to attract electrons towards itself. An atom's electronegativity is affected by both its atomic number and the distance that its valence electrons reside from the charged nucleus...
effects in a chemically intuitive way; - Dipoles, arising from overlap of s and p orbitals and describing lone pairs and other atomic distortions;
- Quadrupoles, arising from the overlap of p orbitals, and associated with pi bonds, for example;
- Octopoles and hexadecapoles can be included if very high accuracy is required.
The DMA describes the potential at points outside the molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
with an accuracy which is essentially that of the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
, so that its use entails no loss of precision. The DMA description gives the electrostatic energy of interaction between two molecules. It does not account for charge overlap effects and hence excludes the penetration energy.
DMA is inherently much more accurate than the commonly used partial charge
Partial charge
A partial charge is a charge with an absolute value of less than one elementary charge unit .-Partial atomic charges:...
methodologies for calculating intermolecular interaction energies, since it captures anisotropy
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
of the atom-atom contributions to electrostatic interaction. It may therefore seem surprising that it has not been more widely used in molecular simulation. Possible reasons for this are:
- Its non-inclusion in popular simulation codes;
- The need to keep track of the orientation of a local axis system for each molecule;
- The conformation-dependence of the DMA. As a consequence of its accuracy, the DMA captures features of the molecular charge distribution that depend strongly on molecular conformation. Thus, in a DMA-based simulation, the multipoles would have to be recalculated whenever a molecule underwent a conformational change.
DMA has found extensive use in crystal structure prediction
Crystal structure prediction
Crystal structure prediction is the calculation of the crystal structures of solids from first principles. Reliable methods of predicting the crystal structure of a compound, based only on its molecular structure, has been a goal of the physical sciences since the 1950s...
for small organic molecules
Organic compound
An organic compound is any member of a large class of gaseous, liquid, or solid chemical compounds whose molecules contain carbon. For historical reasons discussed below, a few types of carbon-containing compounds such as carbides, carbonates, simple oxides of carbon, and cyanides, as well as the...
, where significant progress can often be made while using rigid molecular structures.