 |
|---|
 |
|
|---|
|
Note that definitions of many of the acronyms used in this document can be found in the Quantum Chemistry Acronyms Database (Northern Illinois Univ.).
obtained by solving Schrodinger's equation.Schrodinger's equation addresses the following questions:
configuration, conformation, size, shape, etc.
heat of formation, conformational stability,
chemical reactivity, spectral properties, etc.
Quantum mechanics methods are based on the following principles:
Theoretical Background
A basic understanding of how approximate molecular wave equations are constructed and solved is essential to the proper
use of quantum chemical software. Quantum chemistry is highly mathematical in nature, and the language used to describe
quantum chemical methods more often relates to equations than to chemical concepts. Non-specialists who are interested
in using quantum chemical methods as molecular modeling tools can be faced with a considerable learning curve.
This section is designed to serve as a starting point for acquiring a working knowledge of the current generation of quantum chemical methods. The emphasis in this work is on the relationships between important quantum chemical concepts and their mathematical foundations. This presentation is loosely based on Lowe.
Standing Waves in a Clamped String
The properties of waves in a string clamped at both ends (clamped string) are analogous to some of the important
basic quantum mechanical properties of atoms and molecules. Waves can be generated by plucking (adding energy to) such a string:
 |
|---|
By observation, waves in a clamped string must adopt discrete wavelengths because the ends of the string cannot be displaced.
Such waves can be mathematically described by solving the wave equation for
. The one-dimensional form of the wave equation is sufficient
to describe the clamped string model:
The solutions to the wave equation, , are called the wavefunctions.
is proportional to the energy density at any point along
the wave (i.e. the amount of energy from the pluck that is stored in each part of the string).
All solutions to the wave equation must be symmetric. However, there are two possible kinds of wave symmetries. The first type of symmetry is such that the wave and its reflection about the wave coordinate axis are superimposable (symmetric):
 |
|---|
 |
|---|
The second type of symmetry is such that the wave and its reflection are not superimposable (antisymmetric):
 |
|---|
An asymmetric waveform is shown below for comparison purposes:
 |
|---|
.
Einstein showed in 1905 that the energy of a particle is 
,
where m is the particle's relativistic mass.
Combining these two expressions yields the relationship between a photon's momentum and frequency (or wavelength):
As a particle's momentum becomes large (i.e. due to its mass), 
becomes undetectably small.
It is useful to express in terms of energy:
where 
is the Laplacian operator
(
) and
is the wavefunction describing the displacement at any point
along the wave.
Schrodinger substituted the de Broglie wavelength for
to adapt the classical wave equation to particle waves. The Schrodinger equation is given by:
This equation can be rearranged in a series of algebraic steps to a more convenient form:
Substituting H into Schrodinger's equation leads to:
is called an eigenfunction and E an eigenvalue.
where q is the charge on each particle and r is the distance between them.
If q1 and q2 are both + or both -, they repel (V > 0). If q1 and q2 are opposite, they attract (V < 0). The potential energy between an electron (q = -e) and a nucleus (q = +Ze) can be written as:
Substituting for V in the wave equation gives:
V decreases as the electron and nucleus get closer. So, the electron moves faster when it's close to the nucleus, and slower when it's farther away (because energy is interconverted from T into V to maintain constant E).
The second derivative of is the rate of change of slope of
at any given point, which describes the curvature of "wiggliness" of
the function. So, the wigglier is at a given point in space,
the greater is the kinetic energy of the electron at that point.
and E and its corresponding energy relate to a single electron
bound to a nucleus of charge +Ze. The are called one-electron
orbitals (often called hydrogen-like atomic orbitals). They are designated as 1s, 2s, 2p, etc.
gives the probability density of finding an electron at a given point in space. Each
can be contoured (i.e. graphed at a constant value).
Note that each solution has a radial term containing r, and some have angular terms containing
and 
. The s orbitals are spherical, and therefore have no angular dependence. All other orbitals have spatial directionality, and the corresponding wavefunctions have angular terms.
Under quantum mechanics, however, electrons can exist at larger distances than allowed by V (i.e. E < V).
tapers off exponentially outside of this range (i.e.
contains an imaginary component). This is called "tunneling".
and all chemical properties derived therefrom
are real valued, and so is given by
, where
is the complex conjugate of
.
where dv is an infinitesimal chunk of volume and the c's are normalization constants that adjust the
to satisfy the normalization equation.
can be symmetric or antisymmetric,
but never unsymmetric. The superposition of all wavefunctions of an atom is spherically symmetric (i.e. there is no favored
direction in an atom).
and 
) called "spin" states.
The Schrodinger equation for a multi-electron atom can be solved numerically, although Velectron-electron cannot be
included as an explicit term in the Hamiltonian. Rather, its effect on
can be accounted for by a mathematically simpler approach: that each electron interacts with an average of the nucleus + all
other electrons (self-consistent field approximation).
where h(i) are one-electron Hamiltonians.
Schrodinger's equation can then be approximated as:
where 
and
are the one-electron energies such that
(the total energy)
and 
is composed of a combination of one-electron wavefunctions,
.
Since electron-electron interactions are neglected in this approach, the probability of finding either electron at a given position
(i.e. 
) does not depend on the position of the other
electron. This can be mathematically represented using the products of the one-electron wavefunctions:
The arguments of the wavefunctions denote the positions of each electron.
The multi-electron wavefunction must take into consideration the fact that electrons are indistinguishable, and therefore interchanging electron position assignments in a wavefunction cannot lead to a different wavefunction. This is not the case for the products themselves:
but is true of the sum of permuted products:
and their difference:
where 
and
are the one-electron wavefunctions (e.g. 1s,
2s, etc.).
Interchanging electrons in the two equations leads to:
and
Note that the sign changes for 
when electron position
assignments are interchanged, but the equations are otherwise unaffected by this operation.
and
are called "space" functions because they depend on the spatial
positions of the electrons.
and . Spin states are mathematically represented by "spin" functions, 
and 
, analogous to and .Spin functions must be symmetric or antisymmetric with respect to the interchange of electron state assignments. For a two electron system, the possible spin functions are:
, is composed of a space function
( or
) and a spin function
( or
):
In practice, only the antisymmetric form, 
, is physically
meaningful. results from symmetrically opposite
constituent space and spin functions:
and
There are four possible antisymmetric atomic spin orbitals for a 1s(1) 2s(2) configuration:
are approximate wavefunctions for the entire atom,
made by "mixing" one-electron spin orbitals.Antisymmetric spin orbitals are often constructed using a mathematical function called a Slater determinant.
Electron Correlation Effects
Velectron-electron was neglected in the Hamiltonian in the previous sections. The effects of electron-electron
interactions are, in general, called "electron correlation". and
E cannot be used to correctly predict atomic properties without somehow accounting for electron correlation.
Multi-electron wavefunctions () are overly influenced by
nuclear-electron attraction, and this tends to spatially contract the electron density distribution toward the nucleus. This is
contrary to the unaccounted effect of electron-electron repulsion, which tends to make the orbitals larger and more diffuse.
The overestimated degree of contraction can be corrected by implicitly reducing the effects of nuclear-electron attraction. The rationale is that each electron is screened from the nucleus by the other electrons. The screening effect is assumed to be averaged over all other electrons.
The one-electron wavefunctions () can be improved in this regard by
modifying the nuclear charge constant, Z:
can be written as:
where 
is an adjustable parameter.
As becomes smaller, the orbital tends to expand.
can be determined using the self-consistent field approach (SCF).
In practice, more useful forms of are available
(e.g. Slater and Gaussian-type orbitals).
Note that Tnucleus is omitted. This (Born-Oppenheimer) approximation considers the nuclei to be stationary relative to the electron motions. Since the positions of the nuclei are fixed during the calculations, Vnucleus-nucleus is typically treated as a constant.
Just as for multi-electron atoms, approximate molecular wavefunctions (molecular spin orbitals) can be created using a set of one-electron wavefunctions.
The major differences between the various molecular orbital calculation methods pertains to their consideration (or lack of consideration) of electron correlation.
Instead, wavefunctions are used in which the radial terms, 
,
are simplified. The most common such wavefunctions are Slater-type orbitals (STO):
where s is a screening constant, and Z - s can be taken as
and Gaussian-type orbitals (GTO):
where is a curve-fitting constant used to approximate an STO.
The STO screening constants are calculated for small model molecules using rigorous self-consistent field methods, and then generalized for use with actual molecules of interest. This data is supplied with the various software implementations that use STOs. The mathematical requirements for solving the integrals of the wave equation using STOs are time consuming.
The accuracy of STOs can be improved by combining two or more STOs (i.e. with two different values of
) into a single one-electron wavefunction
(double- basis set):
 |
|---|
Likewise, STOs can be devised that reflect the shape properties of polarized one-electron orbitals (e.g. the combination of a 1s and a 2pz orbital).
Gaussian orbitals are mathematically simpler than STOs, but less accurate. A 1s GTO and the corresponding one-electron hydrogen-like orbital are compared in the following plot:
 |
|---|
All of the one-electron orbitals can be built by combining sets of gaussian functions (gaussian primitives) that approximate each STO. The result is called a contracted gaussian function.
A minimal basis set is one in which only occupied orbitals of each isolated atom are used to compose the molecular orbitals. Unoccupied molecular orbitals are called virtual orbitals. Additional information on basis sets is available (Jan Labanowski, Ohio Supercomputer Center). A library of basis sets is maintained by the Environmental Molecular Sciences Laboratory (Pacific Northwest Laboratory, Battelle Memorial Institute).
be the molecular wavefunctions. be the molecular spin orbitals (analogous to atomic spin
orbitals). be the one-electron wavefunctions used to create the
.
) can be composed of a set of one-electron
orbitals () centered on each nucleus:
where 
are numerical coefficients that determine the amount of
each one-electron wavefunction in each molecular orbital, and m is the size of the basis set.
The can be calculated using various approaches, all of which
are based on the method of linear variation.
The molecular spin orbitals, , can be constructed using:
or
or a Slater determinant.
that best
approximate the actual wavefunction 
. According to the variational
theorem, this is equivalent to finding the that give rise to the
lowest possible energy (minimum energy) for a given choice of Hamiltonian and basis set.
Given a means of evaluating the energy of as a function of
, the energy can be mininimized with respect to
. The average energy is obtained from the following expression:
where 
is an infinitesimal chunk of space and spin.
Substituting the LCAO approximation for in the previous equation
leads to:
The minimum energy with respect to each results when the derivative
of < E > is zero:
Differentiating the equation for < E > with respect to the LCAO coefficients for
leads to:
The integrals are usually abbreviated as:
and
can be rewritten using these abbreviations:
where n is the number of one-electron wavefunctions in the basis set of
.
The preceding set of equations (one for each ) can be used to
solve for < E > via the mathematical method of determinants. The determinant (called the secular determinant)
for the simple case of 
(one electron) is written as:
Solving the secular determinant for < E > requires values for each of the overlap and Coulomb integrals, which can be obtained completely or partially from experimental data (semiempirical methods) or from full evaluation of these integrals (ab initio).
Lastly, the value of < E > can be substituted into the set of:
to solve for the . Note that
(normalization condition) is also required to
solve for . There is one set of
for each member of the basis set.
In practice, this set of equations is expressed in the form of a matrix equation:
where H, S, C, and E are matrices of Coulomb integrals, overlap integrals, the
, and the < E >, respectively.
The solution of HC = SCE for leads to two
molecular orbitals (plotted as three-dimensional contours at a constant value of
):
 |
|---|
Notice that 
is an antibonding molecular orbital due to the fact
that the two lobes of the orbital have no overlap.
A self-consistent field approach to treating electron correlation was previously described in terms of an adjustable parameter,
, in place of the nuclear charge, Z, in the one-electron
wavefunction:
represents the effective nuclear charge interacting with one
electron due to the presence of another. Linear variation can be used to find a single value of
that accounts for the mutual shielding of each electron
(test electron) by an average of all other electrons (average set). This can be done using the variation method, but the process
is iterative. Each electron takes a turn as the test electron and as a member of the average set. However, returning the test
electron to the average set (to consider the next test electron) changes the average shielding. Eventually,
approaches a constant value when each test electron no longer
changes the average shielding effect.
The HF approach considers electron correlation in a similar manner. The Hamiltonian considers each electron, i, in the average field of all other electrons in the molecule (i.e. a two-electron system). The Hamiltonian that describes this approximation (called the Fock operator is given by:
where F is the Fock operator, J (called the Coulomb integral) reflects the average interaction potential of electron i due to all other electrons, and K is a second integral (called the exchange integral) having no simple physical interpretation.
Since J and K are themselves functions of the one-electron molecular orbitals, so is
.
The first two terms in the Fock equation can be recognized as 
,
seen previously in the multi-electron Hamiltonian:
The wave equation can be written as:
Since the (the functions being solved for) are part of B,
the solution is obtained iteratively:
are initially guessed, from which
is computed.
, which
is used to compute a new .
becomes constant.
The molecular orbitals (typically spin orbitals) are constructed from a basis set of one-electron atomic orbitals,
. The electrons can be assigned to the molecular orbitals in
various ways (e.g. 1s(1) 1s(2) 2s(3) 2s(4), 1s(1)
1s(2) 2s(3) 2p(4), etc.).
The LCAO method is used to approximate , with the molecular orbital
coefficients, , calculated via linear variation in which
is now calculated using the Fock operator. So, each SCF
iteration involves solving for a new set of , and the process is
repeated until the approach constant values.
The quality of the HF result depends on the size and quality of the basis set. The accuracy of a HF result is, however, limited by the use of average electron correlation effects.
The 
are estimated using linear variation. CI can greatly add to
the computational requirements of solving the molecular wave equation, but can significantly increase the accuracy of the results.
A number of approximate (semiempirical) HF methods have been developed to simplify these integrals. Further information about the semiempirical methods can be obtained in reviews by Zerner and Stewart.
, and
coefficients .
) and the molecular orbital energies
for all molecular orbitals in a molecule.