The Born-Oppenheimer Approximation

The total molecular wavefunction P(R,r) depends on both the positions of all of the nuclei and the positions of all of the electrons. Since electrons are much lighter than nuclei, and therefore move much more rapidly, electrons can essentially instantaneously respond to any changes in the relative positions of the nuclei. This allows for the separation of the nuclear variables from the electron variables. 

This separation of the total wavefunction into an electronic wavefunction y/(r) and a nuclear wavefunction 0(R) means that the positions of the nuclei can be fixed, leaving it only necessary to solve for the electronic part. This approximation was proposed by Bom and Oppenheimer and is valid for the vast majority of organic molecules. 

The potential energy surface (PES) is created by determining the electronic energy of a molecule while varying the positions of its nuclei. It is important to recognize that the concept of the PES relies upon the validity of the Bom-Oppenheimer approximation so that we can talk about transition states and local minima, which are critical points on the PES. Without it, we would have to resort to discussions of probability densities of the nuclear-electron wavefunction. 

The Hamiltonian obtained after applying the Bom-Oppenheimer approximation and neglecting relativity is 

The wavefunction i (r) depends on the coordinates of all of the electrons in the molecule. Hartree proposed the idea, reminiscent of the separation of variables used by Bom and Oppenheimer, that the electronic wavefunction can be separated into a product of functions that depend only on one electron. 

The solution to a Schrodinger equation involving the electronic Hamiltonian, 

By a parametric dependence we mean that, for different arrangements of the nuclei, O iec is a different function of the electronic coordinates. The nuclear 

Equations (2.10) to (2.14) constitute the electronic problem, which is our interest in this book. 

If one has solved the electronic problem, it is subsequently possible to solve for the motion of the nuclei under the same assumptions as used to formulate the electronic problem. As the electrons move much faster than the nuclei, it is a reasonable approximation in (2.2) to replace the electronic coordinates by their average values, averaged over the electronic wave function. This then generates a nuclear Hamiltonian for the motion of the nuclei in the average field of the electrons, 

The total energy tot( A ) provides a potential for nuclear motion. This function constitutes a potential energy surface as shown schematically in Fig. 2.2. Thus the nuclei in the Bom-Oppenheimer approximation move on a potential energy surface obtained by solving the electronic problem. Solu- 

The above problem can be simplified by separating the fast electronic motion from the slow nuclear motion. One first defines an electronic Hamiltonian, also called clamped nucleus Hamiltonian Hgiir R) = Tgiir) + V r R). This electronic Hamiltonian acts in the electronic space and depends parametrically on the nuclear coordinates R, as indicated by the semicolon in the coordinate dependence of the operators. The eigenfunctions and eigenvalues of the associated time-independent Schrodinger equation (TISE) 

Let US, for simplicity, rewrite the nuclear kinetic energy operator in terms of mass-weighted rectilinear coordinates (defined as the rectilinear coordinates multiplied by the square root of the mass of the nucleus) The first term of the 

Inserting Eq. (2.9) in Eq. (2.8), multiplying from the left by 4 l (r R) and integrating over the electronic coordinates, one obtains 

This last equation shows that the nuclear motion of the molecule obeys an infinite set of coupled differential equations. The so-called non-adiabatic couplings A describe the dynamical interaction between the electronic and nuclear motions [1-3]. They are given by 

The Bom-Oppenheimer approximation [4, 5] consists in neglecting the non-adiabatic couplings. This approximation relies on the very different masses of the electron and nuclei. Indeed, the proton, which is the lightest atomic nucleus, is roughly 1836 times heavier than the electron. Therefore the electron velocity is much higher than that of the nuclei and the fast electrons adjust instantaneously to the slow motion of the nuclei. Within this approximation, no transition between different adiabatic electronic states can be induced by the nuclear motion and in this case, the total molecular wavefunction can be written as 

The Hamiltonian for a molecule is easily determined. The Hamiltonian will include kinetic energy terms for the nuclei (indexed by A) and electrons (indexed by a), electron-nucleus potential (distance of separation ta,), nuclear-nuclear potential (distance of separation of Rab), and electron-electron repulsion (distance of sq aration Tab). 

Though the complete Hamiltonian for a molecule is easily determined, the resulting Schroedinger equation is impossible to solve, even analytically. 

An approximation that can be made is to realize that the motion of the nuclei is sluggish relative to the motion of the electrons due to the large differences in mass. Due to the great difference in motion between the nuclei and the electrons, the electrons are capable of instantaneously adjusting to any change in position of the nuclei. Hence, the electron motion is determined for a fixed position of the nuclei making the distances Rab in Equation 9-1 now a constant. This approximation is called the Born-Oppenheimer approximation. The Bom-Oppenheimer approximation removes the kinetic energy operators for the nuclear motion in Equation 9-1. 

A 1 electrons. nuclei eieetrons Z. ffttelel nuclei Z, Z electrons dectrons 1 

The Schroedinger equation that is solved for then just becomes the electronic Schroedinger equation for the molecule plus a constant term for the nuclear repulsion. 

We now begin the study of molecular quantum mechanics. If we assume the nuclei and electrons to be point masses and neglect spin-orbit and other relativistic interactions (Sections 11.6 and 11.7), then the molecular Hamiltonian is 

Z and Zj3. The fourth terra is the potentid energy of the attractions between the electrons and the nuclei, being the distance between electron i and nucleus a. The last terra is the potential energy of the repulsions between the electrons, r/j being the distance between electrons i and /. The zero level of potential energy for (13.1) corresponds to having all the charges (electrons and nuclei) infinitely far from one another. 

The wave functions and energies of a molecule are found from the Schrddinger equation  

The electronic Hamiltonian including nuclear repulsion is + V n. The nuclerir-repulsion term is given by 

The variables in the electronic Schrddinger equation (13.4) are the electronic coordinates. The quantity V n is independent of these coordinates and is a constant for a given nuclear configuration. Now it is easily proved (Problem 4.49) that the omission of a constant term C from the Hamiltonian does not affect the wave functions and simply decreases each energy eigenvalue by C. Hence, if Vjva/ is omitted from (13.4), we get 

The electronic Hamiltonian including nnclear repnlsion is H i + Vnn- The nuclear-repulsion term Vm is 

The energy U in (13.4) is the electronic energy including internuclear repulsion. The 

if p(r, u. uN) is the electron density at r corresponding to the nuclear geometry u . .., uN, the time-averaged electron density is 

When the electrons can be assigned to specific nuclei, and follow these nuclei perfectly, the density for such a rigid group can be written as 

The label of rigid group as used here may apply to atoms or rigidly connected groups of atoms. In the former case, we obtain 

Expression (2.14), referred to as the convolution approximation, is widely applied in crystallographic work. 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving 

For fixed nuclei = 0, and F is constant, and there is a set of electronic wave functions J/g which satisfy the Schrddinger equation 

The Bom-Oppenheimer approximation is valid because the electrons adjust instantaneously to any nuclear motion they are said to follow the nuclei. For this reason Eg can be treated as part of the potential field in which the nuclei move, so that 

It follows from the Bom-Oppenheimer approximation that the total wave function ij/ can be factorized  

The wave function can be factorized further into a vibrational part ij/ and a rotational part ij//. 

With this approximation the total wavefunction for the molecule-ion can be written as the product of an electronic wavefunction, y/, which is a function only of the electron coordinates, and a nuclear wavefunction, y/, which is a function only of the nuclear coordinates  

The only variables in this Hamiltonian are the coordinates of the electron, which appear in and V -. Although the internuclear separation, R, is treated as a constant when the electronic Schrodinger equation is being solved, the wavefunction obtained will depend upon the value of R used. 

Exercise 2.1 Derive Eq. (2.6) for the time-dependent phase factor. 

The time dependence of the time-dependent wavefunction in Eq. (2.4) corresponds therefore simply to a rotation in the complex plane and the probability density of the time-dependent wavefunction, ( Ek, given as 

The masses of the nuclei, m, are at least three orders of magnitude larger than the mass of an electron. We can therefore assume that the electrons will instantaneously adjust to a change in the positions of the nuclei and that we can find a wavefunction for the electrons for each arrangement of nuclei. In the Born-Oppenheimer approximation the total molecular Hamilton operator Hnuc,e from Eq. (2.1) is thus partitioned in the kinetic energy operator of the nuclei, field free electronic Hamiltonian defined as 

Furthermore, setting up a time-independent Schrodinger equation with this operator we obtain the time-independent field-free electronic Schrodinger equation for a given set of nuclear coordinates Rk  

The solution Rk ) of this equation is called the electronic wavefunction 

The Born-Oppenheimer Approximation.—The total wavefunction for a molecule may be approximated by separation into two factors, 

Here V(R,r) symbolizes the potential energy of electrons in the field of nuclei whose co-ordinates are given by R. This equation may be solved for different values of R, since the hamiltonian contains no differential operators with respect to R. The eigenvalues plotted against R give the potential-energy curve or surface referred to above, f nuciear is then a solution of the nuclear Schrodinger equation, 

Bom and Oppenheimer13 showed that this approximation is equivalent to the neglect of terms in the hamiltonian of the order of 1/Af, i.e. about 10 3 even for the lightest nuclei. 

In the absence of external fields, we may take axes that move laterally with the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass ti=MaMjs/(Ma+Mb), moving in a spherically symmetrical potential U(R), where R is the intranuclear separation. The Schrodinger equation is therefore 

This equation may be separated analogously to that for the hydrogen atom, the angular functions being spherical harmonics 

The Born-Oppenheimer approximation exploits the fact that the nuclear mass is very much larger than the electronic mass, and therefore the nuclear dynamics are expected to be slow in comparison to the electronic dynamics. Thus, it is convenient to introduce an electronic state, i R ), that is determined by a set of static nuclear coordinates, R. i R ) thus depends parametrically on R.  

The Born-Oppenheimer approximation is to assume that a many body state, /), may be factorized as a single, direct product of an electronic state, i R ), and a nuclear state vi) associated with the electronic state, i R )  

Then /) satisfies an eigenvalue equation and i R ) is an eigenstate of the so-called Born-Oppenheimer Hamiltonian, provided that that the electronic state r, R ) is so weakly parametrized by the nuclear coordinates that. 

By definition, f R ) is an eigenstate of Hbo whose corresponding eigenvalue, Pj( R ), is the sum of the electronic kinetic energy and all the potential energy terms. Pj( R ) is also known as the adiabatic potential energy surface, and is shown schematically in Fig. 2.1. As we shall see shortly, Pj( R ) is the effective potential experienced by the nuclei. 

The Born-Oppenheimer approximation is an adiabatic approximation, as it is equivalent to the assumption that there are no transitions between the electronic 

This calculation assumes, however, that the internuclear distance in H2 is fixed, the molecule behaving like two balls linked by a rigid rod. This in fact is not the case the system resembles two balls linked by a spring rather than 

The essence of the Bom-Oppenheimer approximation is that the electronic and nuclear motions (following charge transfer) are separable in time. This can be understood tentatively, if one considers that the (CN) ligand is about 50 000 times heavier than an electron. It follows that the charge-transfer step presented by Eq. (5,37) should, in fact, be written in two steps, namely 

Transfer of the electron takes place without movement of the nuclei. The total energy of the system is increased (because the species formed in Eq. (5.38) is not in its most stable state). The energy released when the product of charge transfer decays to its equilibrium state is referred to as the solvent reorganization energy, X, which is of central importance in the Marcus theory of charge transfer. 

On the basis of the above physical model, we show the residts of the Marcus theory, without showing the detailed calculations, which can be found in most textbooks of electrochemistry. The standard Gibbs energy of activation for an outer-sphere charge-transfer process is given approximately by 

It is not easy to calculate or experimentally determine the value of X. Estimated values given in the literature for aqueous solutions are in the range of approximately 50-200 kj mol h For reactions taking place in a solvent oflower polarity, this number could be much smaller, of course. 

Born and Oppenheimer showed in 1927 [9] that to a very good approximation the nuclei in a molecule are stationary with respect to the electrons. This is a qualitative expression of the principle mathematically, the approximation states that the Schrodinger equation (chapter 4) for a molecule may be separated into an electronic and a nuclear equation. One consequence of this is that all ( ) we have to do to calculate the energy of a molecule is to solve the electronic Schrodinger equation and then add the electronic energy to the internuclear repulsion (this latter quantity is trivial to calculate) to get the total internal energy (see section 4.4.1). A deeper consequence of the Born-Oppenheimer approximation is that a molecule has a shape. 

The nuclei see the electrons as a smeared-out cloud of negative charge which binds them in fixed relative positions (because of the mutual attraction between electrons and nuclei in the internuclear region) and which defines the (somewhat fuzzy) surface [10] of the molecule (see Fig. 2.11). Because of the rapid motion of the electrons compared to the nuclei the permanent geometric parameters of the molecule are the nuclear coordinates. The energy (and the other properties) of a molecule is a function of the electron coordinates ( = (x, y, z of each electron) section 5.2), but depends only parametrically on the nuclear coordinates, i.e. for each geometry 1,2. there is a particular energy i = (x, y, z.), 2 = y,z.) cf. x , which 

Consider the molecule H3, made up of three protons and two electrons. Ab initio calculations assign it the geometry shown in Fig. 2.12. The equilibrium positions of the nuclei (the protons) lie at the comers of an equilateral triangle and has a definite shape. But suppose the protons were replaced by positrons, which have the same mass as electrons. The distinction between nuclei and electrons, which in molecules rests on mass and not on some kind of charge chauvinism, would vanish. We would have a quivering cloud of flitting particles to which a shape could not be assigned on a macroscopic time scale. 

A calculated PES, which we might call a Bom-Oppenheimer surface, is normally the set of points representing the geometries, and the corresponding energies, of a collection of atomic nuclei the electrons are taken into account in the calculations as needed to assign charge and multiplicity (multiplicity is connected with the number of unpaired electrons). Each point corresponds to a set of stationary nuclei, and in this sense the surface is somewhat unrealistic (see section 2.5). 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the 

Amolecular system in HyperChem consists of N nuclei A (having positive charges, -t-Z ) and M electrons i (each with negative charge, q = -1). The nuclei are described by a vector, R, with 3N Cartesian X, Y, and Z components. The electrons are described by a vector, r, with 3M Cartesian x, y, and z components. The electrons are explicitly considered only in semi-empirical calculations. 

A Hamiltonian is the quantum mechanical description of an energy contribution. The exact Hamiltonian for a molecular system is  

Since nuclei are much heavier than electrons and move slower, the Born-Oppenheimer Approximation suggests that nuclei are stationary and thus that we can solve for the motion of electrons only. This leads to the concept of an electronic Hamiltonian, describing the motion of electrons in the potential of fixed nuclei. 

HyperChem s semi-empirical calculations solve (approximately) the Schrodinger equation for this electronic Hamiltonian leading to an electronic wave function I eiecW for the electrons  

The first step of the adiabatic description is the Born-Oppenheimer approximation, according to which 

The adiabatic approximation is based on the fact that typical electronic velocities are much greater than typical nuclear (ionic) velocities. (The significant electronic velocity is v = 10 cm/s, whereas typical nuclear velocities are at most of order 10 cm/s.) One therefore assumes that, because the nuclei have much lower velocities than the electrons, at any moment the electrons will be in their ground state for that particular instantaneous nuclear configuration. 

Under circumstances where TN(q) = 0, and at particular arrangement of the ion cores, we can separate electronic and nuclear motions. This can be accomplished by selecting some basis set of electronic wavefunctions p (r q), which satisfy the partial Schrbdinger equation 

In deducing this result, we have used Equation 1.5 and the fact that the wavefiinction (pj, is an eigenfunction of Equation 1.5. Multiplying from the left by p and integrating over the electronic coordinates, we obtain the usual set of coupled equations for the Xav [4, 5] (see also Ref [6] with modifications given by McLachlan [7] and Kolos [8])  

The restriction fo / a in Equation 1.8a is a consequence of the orthonormality of the cpj, (Pi, (Pa)r = bai)- Here and in Equation 1.8a, angle brackets indicate integration over the electronic coordinates only. To avoid confusion resulting from numerous subscripts, it is often convenient to adopt a matrix notation, writing Equation 1.8a as 

The first thing to note is that the nuclei are very much more massive than the electron (by a factor of 1836). If they were classical particles, we might argue that their velocities would be very much less than the velocity of the electron, and so to a first approximation the motion of the electron should be the same as if the nuclei were fixed in space. 

If the motions of the electron and of the two nuclei are indeed independent of one another, the total wavefunction should be a product of an electronic one and a nuclear one, 

We might reasonably expect that the electronic wavefunction would depend on the particular values of RA and RB at which the nuclei were fixed, and I have indicated this in the expression above. 

Bom and Oppenheimer tackled the problem quantum-mechanically in 1927 their treatment is pretty involved, but the basic physical idea is as outlined above. To simplify the notation, I will write the total Hamiltonian as follows  

Had we been dealing with a polyelectron system, there would have been extra terms in the total Hamiltonian to take account of the electron-electron repulsion. These would have also been collected into Ht. 

The exact Hamiltonian H for a diatomic molecule, with the electronic coordinates expressed in the molecule-fixed axis system, is rather difficult to derive. Bunker (1968) provides a detailed derivation as well as a review of the coordinate conventions, implicit approximations, and errors in previous discussions of the exact diatomic molecule Hamiltonian. 

Our goal is to find the exact solutions, ipf (T = Total), of the Schrodinger equation, 

To solve Eq. (3.1.1), it would be useful to write the total energy as a sum of contributions from interactions between different particles. In decreasing order of importance, there are electronic energy, Eel, vibrational energy, G(v), and rotational energy, F(J). In fact, this separation is assumed whenever the expression 

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion terms, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Bom-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

3Michael Polanyi, Hungarian-British chemist, economist, and philosopher. Bom Budapest 1891. Doctor of medicine 1913, Ph.D. University of Budapest, 1917. Researcher Kaiser-Wilhelm Institute, Berlin, 1920-1933. Professor of chemistry, Manchester, 1933-1948 of social studies, Manchester, 1948-1958. Professor Oxford, 1958-1976. Best known for book Personal Knowledge , 1958. Died Northampton, England, 1976. 

4Max Bom, German-British physicist. Bom in Breslau (now Wroclaw, Poland), 1882, died in Gottingen, 1970. Professor Berlin, Cambridge, Edinburgh. Nobel Prize, 1954. One of the founders of quantum mechanics, originator of the probability interpretation of the (square of the) wave-function (Chapter 4). 

Robert Oppenheimer, American physicist. Bom in New York, 1904, died in Princeton 1967. Professor California Institute of Technology. Fermi award for nuclear research, 1963. Important contributions to nuclear physics. Director of the Manhattan Project 1943-1945. Victimized as a security risk by senator Joseph McCarthy s Un-American Activities Committee in 1954. Central figure of the eponymous PBS TV series (Oppenheimer Sam Waterston). 

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Pack R T and Hirschfelder J O 1970 Energy corrections to the Born-Oppenheimer approximation. The best adiabatic approximation J. Chem. Phys. 52 521-34... 

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

The proper quantumdynamical treatment of fast electronic transfer reactions and reactions involving electronically excited states is very complex, not only because the Born-Oppenheimer approximation brakes down but... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Combes, J. M. The Born-Oppenheimer approximation. Acta Phys. Austriaca 17 (1977) Suppl. 139-159... 

Finally, we like to mention that the QCMD model reduces to the Born-Oppenheimer approximation in case the ratio of the mass m of the quantum particles to the masses of the classical particles vanishes [6], This implies... 

Both inoleciilar and qiiantnin mechanics in ethods rely on the Born-Oppenheimer approximation. In qnantiinn mechanics, the Schrddmger equation (1) gives the wave function s and energies of a inolecii le. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

Vhen calculating the total energy of the system, we should not forget the Coulomb inter-ction between the nuclei this is constant within the Born-Oppenheimer approximation Dr a given spatial arrangement of nuclei. When it is desired to change the nuclear positions,... 

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

The mathematical definition of the Born-Oppenheimer approximation implies following adiabatic surfaces. However, software algorithms using this approximation do not necessarily do so. The approximation does not reflect physical reality when the molecule undergoes nonradiative transitions or two... 

The total energy in an Molecular Orbital calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system. This is the potential energy for nuclear motion in the Born-Oppenheimer approximation (see page 32). 

Within the Born-Oppenheimer approximation discussed earlier, you can solve an electronic Schrodinger equation... 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

There are phenomena such as the Renner and the Jahn-Teller effects where the Bom-Oppenheimer approximation breaks down, hut for the vast majority of chemical applications the Born-Oppenheimer approximation is a vital one. It has a great conceptual importance in chemistry without it we could not speak of a molecular geometry. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

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