In
atomic, molecular, and optical physicsAtomic, 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 three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of the...
as well as in
quantum chemistryQuantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. The description of the electronic behavior of atoms and molecules as pertaining to their reactivity is one of the applications of quantum...
,
molecular Hamiltonian is the name given to the
HamiltonianIn quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
representing the
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
of the
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s and
nucleiThe nucleus is the very dense region consisting of nucleons at the center of an atom. Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus, with a very small contribution from the orbiting electrons....
in a
moleculeA molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense...
. This
Hermitian operatorIn mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
and the associated
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
play a central role in
computational chemistryComputational chemistry is a branch of chemistry that uses computers 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...
and
physicsComputational physics is the study and implementation of numerical algorithms to solve problems in physics for which a quantitative theory already exists...
for computing properties of molecules and aggregates of molecules, such as
conductivityConductivity may refer to:*Electrical conductivity, a measure of a material's ability to conduct an electric current*Conductivity , a measurement of an electrolytic solution, such as water...
,
opticalOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, and
magnetic propertiesIn physics, the term magnetism is used to describe how materials respond on the microscopic level to an applied magnetic field; to categorize the magnetic phase of a material. For example, the most well known form of magnetism is ferromagnetism such that some ferromagnetic materials produce their...
, and
reactivityReactivity refers to the rate at which a chemical substance tends to undergo a chemical reaction in time. In pure compounds, reactivity is regulated by the physical properties of the sample. For instance, grinding a sample to a higher specific surface area increases its reactivity. In impure...
.
The bricks of the molecule are the nuclei, characterized by their
atomic numberIn chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
,
Z, while the electrons, which have negative
elementary chargeThe elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. This is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the "elementary positive...
,
q, are its mortar. The charge of a nucleus is
Zq. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the
kinetic energieThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
s of the electrons and the
Coulomb (electrostatic) interactionsCoulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was studied and first published in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the
Coulomb Hamiltonian. From it are missing a number of small terms, most of which are due to electronic and nuclear
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
.
Although it is generally assumed that the solution of the time-independent
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like the hydrogen molecule.
Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by
Born and OppenheimerIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
. The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei
enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called
clamped nucleus Hamiltonian, also called
electronic Hamiltonian, that acts only on functions of the electronic coordinates.
Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate eigenvalue (usually the lowest) can be seen as a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
of the nuclear coordinates, which leads to a
potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...
. In practical calculations the surface is usually
fitThe method of least squares is applied to approximate solutions of overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis....
ted in terms of some analytic functions. In the second step of the
Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the
Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this
articleIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
for more details on this.
The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory)
frameA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, but then the
translationIn physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
al and
rotationA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
al (external) energies are not accounted for. Only the (internal) atomic
vibrationVibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
s enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a
quadratic functionA quadratic function, in mathematics, is a polynomial function of the formThe graph of a quadratic function is a parabola whose major axis is parallel to the y-axis....
of the atomic displacements. This gives the
harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:where k is a positive constant....
Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.
Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an
Eckart frameThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for
rotationA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
,
translationTranslation is the interpreting of the meaning of a text and the subsequent production of an equivalent text, likewise called a "translation," that communicates the same message in another language...
and
vibrationVibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as
Watson's nuclear motion Hamiltonian, but it is also known as the
Eckart Hamiltonian,
Coulomb Hamiltonian
The algebraic form of many observables—i.e., Hermitian operators representing observable quantities—is obtained by the following quantization rules:
- Write the classical form of the observable in Hamilton form (as a function of momenta p and positions q). Both vectors are expressed with respect to an arbitrary inertial frame, usually referred to as laboratory-frame or space-fixed frame.
- Replace p by and interpret q as a multiplicative operator. Here is the nabla
In vector calculus, del is a vector differential operator represented by the nabla symbol: .Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember...
operator, a vector operator consisting of first derivatives. The well-known commutation relations for the p and q operators follow directly from the differentiation rules.
Classically the electrons and nuclei in a molecule have kinetic energy of the form
p2/
(2m) and
interact via
Coulomb interactionsCoulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was studied and first published in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
, which are inversely proportional to the distance
rij
between particle
i and
j.
In this expression
ri stands for the coordinate vector of any particle (electron or nucleus). But from here on we will reserve capital
R to represent the nuclear coordinate, and lower case
r for the electrons of the system. The coordinates can be taken to be expressed with respect to any Cartesian frame centered anywhere in space, because distance, being an inner product, is invariant under rotation of the frame and, being the norm of a difference vector, distance is invariant under translation of the frame as well.
By quantizing the classical energy in Hamilton form one obtains the
a molecular Hamilton operator that is often referred to as the
Coulomb Hamiltonian.
This 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
At a point in space, the electric potential is potential energy divided by charge that is associated with a static electric field. It is a scalar quantity, typically measured in volts....
for more details.
Here
Mi is the mass of nucleus
i,
Zi is the
atomic numberIn chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
of nucleus
i, and
me is the mass of the electron. The
Laplace operatorIn mathematics and physics, the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. It is denoted by the symbols Δ, ∇2, or ∇·∇. In physics, it is used in the...
of particle
i is :. Since the kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to which
xi,
yi, and
zi are expressed. The kinetic energy operator, however, is not invariant under translation (choice of origin of the frame).
Small terms
In the 1920s much spectroscopic evidence made it clear that the Coulomb Hamiltonian
is missing certain terms. Especially for molecules containing heavier atoms, these terms, although much smaller than kinetic and Coulomb energies, are nonnegligible. These spectroscopic observations led to the introduction of a new degree of freedom for electrons and nuclei, namely
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
. This empirical concept was given a theoretical basis by
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
when he introduced a relativistically correct (Lorentz covariant) form of the one-particle Schrödinger equation. The Dirac equation predicts that spin and spatial motion of a particle interact via
spin-orbit couplingIn quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion...
. In analogy spin-other-orbit coupling was introduced. The fact that particle spin has some of the characteristics of a magnetic dipole led to
spin-spin couplingMagnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. The energy of the interaction is as follows:...
. Further terms without a classical counterpart are the Fermi-contact term (interaction of electronic
density on a finite size nucleus with the nucleus), and nuclear quadrupole coupling (interaction of a nuclear
quadrupoleA 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...
with the gradient of an electric field due to the electrons). Finally a parity violating term predicted by the
Standard ModelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
must be mentioned. Although it is an extremely small interaction, it has attracted a fair amount of attention in the scientific literature because it gives different energies for the enantiomers in chiral molecules.
The remaining part of this article will ignore spin terms and consider the solution of the eigenvalue (time-independent Schrödinger) equation of the Coulomb Hamiltonian.
The Schrödinger equation of the Coulomb Hamiltonian
The Coulomb Hamiltonian has a continuous spectrum due to the center of mass motion of the molecule in homogeneous space. In classical mechanics it is easy to separate off the
center of massThe center of mass of a system of particles is a specific point where, for many purposes, the system behaves as if its mass were concentrated there. The center of mass is a function only of the positions and masses of the particles that compose the system...
(COM) motion of a system of point masses. Classically the motion of the COM is uncoupled from the other motions. The COM moves uniformly (i.e., with constant velocity) through space as if it were a point particle with mass equal to the sum
Mtot of the masses of all the particles.
In quantum mechanics a free particle has as state function a plane wave function, which is a non-square-integrable function of well-defined momentum. The kinetic energy
of this particle covers . The position of the COM is uniformly probable everywhere, in agreement with the Heisenberg uncertainty principle.
Quantum mechanically the proper separation of the COM motion
is much more cumbersome than in classical mechanics. By introducing the coordinate vector of the center of mass as three of the degrees of freedom of the system and eliminating the coordinate vector of one (arbitrary) particle, so that the number of degrees of freedom stays the same, one obtains by a linear transformation a new set of coordinates
ti. These coordinates are linear combinations of the old coordinates of
all particles (nuclei
and electrons). By applying the
chain ruleIn calculus, the chain rule is a formula for the derivative of the composite of two functions.In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of y with respect to x can be computed as the rate of change of...
one can show that
The first term of is the kinetic energy of the COM motion, which can be treated separately since does not depend on . As just stated, its eigenstates are plane waves. The constants 1/μ
ij are positive and are linear combinations of all the inverse masses 1/
mi. They are generalized
reduced massReduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
es. The potential consists of the Coulomb terms expressed in the new coordinates. The first term of has the usual appearance of a kinetic energy operator. The second term is known as the
mass polarization term. The translationally invariant Hamiltonian can be shown to be
self-adjointIn mathematics, an element x of a star-algebra is self-adjoint if .A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
and to be bounded from below. That is, its lowest eigenvalue is real and finite. Although is necessarily invariant under permutations of identical particles (since and the COM kinetic energy are invariant), its invariance is not manifest.
Not many actual molecular applications of exist, see, however, the seminal work on the hydrogen molecule for an early application. In the great majority of computations of molecular wavefunctions the electronic
problem is solved with the clamped nucleus Hamiltonian arising in the first step of the
Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
.
See Ref. for a thorough discussion of the mathematical properties of the Coulomb Hamiltonian. Also it is discussed in this paper whether one can arrive
a priori at the concept of a molecule (as a stable system of electrons and nuclei with a well-defined geometry) from the properties of the Coulomb Hamiltonian alone.
Clamped nucleus Hamiltonian
The clamped nucleus Hamiltonian describes the energy of the electrons in the electrostatic field of the nuclei, where the nuclei are assumed to be stationary with respect to an inertial frame.
The form of the electronic Hamiltonian is
The coordinates of electrons and nuclei are expressed with respect to a frame that moves
with the nuclei, so that the nuclei are at rest with respect to this frame. The frame stays parallel to a space-fixed frame. It is an inertial frame because the nuclei are assumed not to be accelerated by external forces or torques. The origin of the frame is arbitrary, it is usually positioned on a central nucleus or in the nuclear center of mass. Sometimes it is stated that the nuclei are "at rest in a space-fixed frame". This statement implies that the nuclei are viewed as classical particles, because a quantum mechanical particle cannot be at rest. (It would mean that it had simultaneously zero momentum and well-defined position, which contradicts Heisenberg's uncertainty principle).
Since the nuclear positions are constants, the electronic kinetic energy operator is invariant under translation over any nuclear vector. The Coulomb potential, depending on difference vectors, is invariant as well. In the description of atomic orbitals and the computation of integrals over atomic orbitals this invariance is used by equipping all atoms in the molecule with their own localized frames parallel to the space-fixed frame.
As explained in the article on the
Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
, a sufficient number of solutions
of the Schrödinger equation of leads to a
potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...
(PES) . It is assumed that the functional dependence of
V on its coordinates is such that
for
where
t and
s are arbitrary vectors and Δφ is an infinitesimal angle,
Δφ >> Δφ
2. This invariance condition on the PES is automatically fulfilled when the PES is expressed in terms of differences of, and angles between, the
Ri, which is usually the case.
Harmonic nuclear motion Hamiltonian
In the remaining part of this article we assume that the molecule is
semi-rigidA semi-rigid molecule is a molecule which has a potential energy surface with a well-defined minimum corresponding to a stable structure of the molecule...
. In the second step of the BO approximation the nuclear kinetic energy
Tn is reintroduced and the Schrödinger equation with Hamiltonian
is considered. One would like to recognize in its solution: the motion of the nuclear center of mass (3 degrees of freedom), the overall rotation of the molecule (3 degrees of freedom), and the nuclear vibrations. In general, this is not possible with the given nuclear kinetic energy, because it does not separate explicitly the 6 external degrees of freedom (overall translation and rotation) from the 3
N-6 internal degrees of freedom. In fact, the kinetic energy operator here
is defined with respect to a space-fixed (SF) frame. If we were to move the origin of the SF frame to the nuclear center of mass, then, by application of the
chain ruleIn calculus, the chain rule is a formula for the derivative of the composite of two functions.In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of y with respect to x can be computed as the rate of change of...
, nuclear mass polarization terms would appear. It is customary to ignore these terms altogether and we will follow this custom.
In order to achieve a separation we must distinguish internal and external coordinates, to which end Eckart introduced
conditionsThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
to be satisfied by the coordinates. We will
show how these conditions arise in a natural way from a harmonic analysis in mass-weighted Cartesian coordinates.
In order to simplify the expression for the kinetic energy we introduce mass-weighted displacement coordinates.
Since
the kinetic energy operator becomes,
If we make a Taylor expansion of
V around the equilibrium geometry,
and truncate after three terms (the so-called harmonic approximation), we can describe
V with only the third term. The term
V0 can be absorbed in the energy (gives a new zero of energy). The second term
is vanishing because of the equilibrium condition.
The remaining term contains the
Hessian matrixIn mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
F of
V, which is symmetric and may be diagonalized with an orthogonal 3
N x 3
N matrix with constant elements:
It can be shown from the invariance of
V under rotation and translation that six of the eigenvectors of
F (last six rows of
Q) have eigenvalue zero (are zero-frequency modes). They span the
external space.
The first 3
N-6 rows of
Q are—for molecules in their ground state—eigenvectors with non-zero eigenvalue; they are the internal
coordinates and form an orthonormal basis for a 3
N - 6 dimensional subspace of
the nuclear configuration space
R3N, the
internal space.
The zero-frequency eigenvectors are orthogonal to the eigenvectors of non-zero frequency.
It can be shown that these orthogonalities are in fact the
Eckart conditionsThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
. The kinetic
energy expressed in the internal coordinates is the internal (vibrational) kinetic energy.
With the introduction of normal coordinates
the vibrational (internal) part of the Hamiltonian for the nuclear motion becomes in the
harmonic approximation
The corresponding Schrödinger equation is easily solved, it factorizes into 3
N-6 equations for one-dimensional
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:where k is a positive constant....
s. The main effort in this approximate solution of the nuclear motion Schrödinger equation is the computation of the Hessian
F of
V and its diagonalization.
This approximation to the nuclear motion problem, described in 3
N mass-weighted Cartesian coordinates, became standard in
quantum chemistryQuantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. The description of the electronic behavior of atoms and molecules as pertaining to their reactivity is one of the applications of quantum...
, since the days (1980s-1990s) that algorithms for accurate computations of the Hessian
F became available. Apart from the harmonic approximation, it has as a further deficiency that the external (rotational and translational) motions of the molecule are not accounted for. They are accounted for in a rovibrational Hamiltonian
that sometimes is called
Watson's Hamiltonian.
Watson's nuclear motion Hamiltonian
In order to obtain a Hamiltonian for external (translation and rotation) motions coupled
to the internal (vibrational) motions, it is common to return at this point to classical mechanics and to formulate the classical kinetic energy corresponding to these motions of the nuclei. Classically it is easy to separate the translational—center of mass—motion from the other motions. However, the separation of the rotational from the vibrational motion is more difficult and is not completely possible. This ro-vibrational separation was first achieved by Eckart in 1935 by imposing by what is now known as
Eckart conditionsThe Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
. Since the problem is described in a frame (an "Eckart" frame) that rotates with the molecule, and hence is a non-inertial frame, energies associated with the
fictitious forceA fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame...
s:
centrifugalIn classical mechanics, centrifugal force is an outward force associated with curved motion, that is, rotation about some center...
and
Coriolis forceIn physics, the Coriolis effect is an apparent deflection of moving objects when they are viewed from a rotating reference frame.Newton's laws of motion govern the motion of an object in an inertial frame of reference. When transforming Newton's laws to a rotating frame of reference, the Coriolis...
appear in the kinetic energy.
In general, the classical kinetic energy
T defines the metric tensor
g = (
gij) associated with the
curvilinear coordinatesCurvilinear coordinates are a coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate system with the same number of coordinates in which the coordinate lines are curved...
s = (
si) through.
The quantization step is the transformation of this classical kinetic energy into a quantum mechanical operator. It is common to follow Podolsky by writing down the
Laplace-Beltrami operatorIn differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces, or more generally on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami operator, after Laplace and...
in the same (generalized, curvilinear) coordinates
s as used for the classical form. The equation for this operator requires the inverse of the metric tensor
g and its determinant. Multiplication of the Laplace-Beltrami operator by gives the required quantum mechanical kinetic energy operator. When we apply this recipe to Cartesian coordinates, which have unit metric, the same kinetic energy is obtained as by application of the quantization rules.
The nuclear motion Hamiltonian was obtained by Wilson and Howard in 1936,
who followed this procedure, and further refined by Darling and Dennison in 1940. It remained the standard until 1968, when Watson was able to simplify it drastically by commuting through the derivatives the determinant of the metric tensor. We will give the ro-vibrational Hamiltonian obtained by Watson,
which often is referred to as the
Watson Hamiltonian. Before we do this we must mention
that a derivation of this Hamiltonian is also possible by starting from the Laplace operator
in Cartesian form, application of coordinate transformations, and use of the chain rule.
The Watson Hamiltonian, describing all motions of the
N nuclei, is
The first term is the center of mass term
The second term is the rotational term akin to the kinetic energy of the
rigid rotorThe rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
. Here
is the α component of the body-fixed
rigid rotor angular momentum operator,
see this article for its expression in terms of
Euler anglesThe Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in 3-dimensional Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles...
. The operator is a component of an operator known
as the
vibrational angular momentum operator (although it does
not satisfy angular momentum commutation relations),
with the
Coriolis coupling constant:
Here ε
αβγ is the
Levi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.-Definition:In three dimensions, the Levi-Civita...
. The terms quadratic in the are centrifugal terms, those bilinear in and are Coriolis terms.
The quantities
Q s, iγ are the components of the normal coordinates introduced above.
Alternatively, normal coordinates may be obtained by application of Wilson's
GF methodThe GF method, sometimes referred to as FG method, is a classical mechanical method introduced by E. Bright Wilson to obtain certain internal coordinates fora vibrating semi-rigid molecule, the so-called normal coordinates Qk...
.
The 3 x 3 symmetric matrix is called the
effective reciprocal inertia tensor. If all
q s were zero (rigid molecule) the Eckart frame would coincide with a principal axes frame (see
rigid rotorThe rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
) and would be diagonal, with the equilibrium reciprocal moments of inertia on the diagonal. If all
q s would be zero, only the kinetic energies of translation and rigid rotation would survive.
The potential-like term
U is the
Watson term:
proportional to the trace of the effective reciprocal inertia tensor.
The fourth term in the Watson Hamiltonian is the kinetic
energy associated with the vibrations of the atoms (nuclei) expressed in normal coordinates
qs, which as stated above, are given in terms of nuclear displacements ρ
iα by
Finally
V is the unexpanded potential energy by definition depending on internal coordinates only. In the harmonic approximation it takes the form
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 calculated at the equilibrium configuration.
This is also called the "Harmonic vibrational and
rigid-rotorThe rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
model."
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
with
with the 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 or vibrations if 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
where refers to the energy of the state . To solve the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
it is needed to decouple the motion of the nuclei and electrons. This is done by approximating the molecular
wavefunctionA wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics describe how the wave function evolves over time...
to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by
where is the electronic and nuclear vibration
quantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
. This formulation is termed an adiabatic wavefunction.
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
where
with the electronic wavefunctions found the nuclear vibrational coordinates or 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 . The electronic wavefunctions defines a complete orthonomal set of functions for each so the molecular wavefunction can be expanded in the basis.
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
where the expansion coefficients describes the vibrational eigenfunctions and the describe the vibrational potential energy. The eigenvalue, 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
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
-
phononIn physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's...
coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as
vibronic couplingIn theoretical chemistry, the vibronic coupling terms, , are proportional to the interaction between electronic and nuclear motions of molecules. The term "vibronic" originates from the concatenation of the terms "vibrational" and "electronic"...
which is important in the case of
avoided crossingAvoided crossing . The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold of N-2 dimensions. In the case of a diatomic molecule , this means that the eigenvalues do not cross...
s or
conical intersectionIn 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 . Conical intersections are ubiquitous in both trivial and non-trivial chemical...
s.
The so-called 'diagonal Born-Oppenheimer correction' (DBOC) can be obtained as
where is the nuclear kinetic energy operator and the electronic wavefunction is parametrically (not explicitly) dependent on the nuclear coordinates.
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See also
- Quantum chemistry computer programs
Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock and some post-Hartree–Fock methods. They may also include density functional theory , molecular mechanics or semi-empirical quantum...
- Adiabatic process (quantum mechanics)
- Franck-Condon principle
The Franck-Condon principle is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions. Vibronic transitions are the simultaneous changes in electronic and vibrational energy levels of a molecule due to the absorption or emission of a photon of the...
- Born-Oppenheimer approximation
In quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons...
- GF method
The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by E. Bright Wilson to obtain certain internal coordinates fora vibrating semi-rigid molecule, the so-called normal coordinates Qk...
- Eckart conditions
The Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external ...
- Rigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...