Born-Oppenheimer, approximation

The first part of the Bom-Oppenheimer (BO) approximation uses the assumption that the nuclei are fixed. The eigenfunctions, then, are purely electronic, but depend 

is the Hamiltonian in Equation 4.1 with the first term neglected I, 2,. .., N stand for the electronic coordinates 1 = (xi,yi,Zi, i), etc. and E IQ) forms a connected surface as a function of Q, called the potential energy surface (PES). By fitting with known mathematical functions, PES may be written with an explicit nuclear coordinate dependence. We will soon show that the PES, that is, Eg(Q), is a potential surface for the motion of the nuclei. 

we include the kinetic energy of the nuclei, and write H of Equation 4.1 in the following way  

The method of the original Born-Oppenheimer approximation involves an expansion about equilibrium positions. Since isolated molecules may be considered to be near equilibrium in many instances, this development is useful in the description of species before or after reaction. The deficiencies in the present approximation are briefly noted in Section II-B (2). 

We may regard Eq. (3) as broken up into a number of equations, one for each of the isolated molecules in the reactant or product phase. Equation (17) below is then representative of one of these equations for one molecule with v nuclei and n electrons  

We want to solve Eq. (17). In the Bom-Oppenheimer approximation one takes advantage of the large discrepancy in the masses, 

let us symbolize the electronic coordinates as x, with 1 3m, and assume a transformation of the nuclear coordinates which divides them into internal (vibrational) coordinates 1, with 1 3v — 6, and external (rotational and trans- 

Before continuing with the general solution we deal with a somewhat fictitious problem. Its solution will, however, shortly prove to have an important hearing on the more reahstic complete solution. If we put x = 0 in Eq. (23), this amoimts to taking the nuclei to be fixed, as they will have no kinetic energy. One can then attempt the solution of 

In the adiabatic approximation, = / H 4 idTe represents a small correction to E (R). Neglecting the correction results in the Bom-Oppenheimer approximation 

Note that in the Bom-Oppenheimer approximation the potential energy for the motion of the nuclei E (R) is independent of the mass of the nuclei, whereas in the adiabatic approximation the potential energy E (R)+H[j(R) depends on the mass. 

Julius Robeit OpjAsiiheimer (1904-1967), American physicist, professor at the University of California in Berkeley and the Califomia Institute of Technology in Pasadena, and at the Institute for Advanced Study in Princeton. In 1943-1945 Oppenheimer headed the Manhattan Project (atomic bomb). 

In order to simplify the Hamiltonian of Eq. (8.66), we freeze the nuclear motion and can then neglect the kinetic energy operators for the nuclei. In this clamped-nuclei approximation the remaining electronic Hamiltonian reads. 

Since the nucleus-nucleus interaction energy operator Vb is a multiplicative constant with respect to integration over electronic coordinates, it can simply be subtracted from the Hamiltonian, 

Therefore, we do not refer to this constant shift of the electronic energy in the following chapters and consider the electronic energy shifted by this term, for which we must, however, keep in mind that it depends on the nuclear coordinates and, hence, on the molecular structure. Note also that the eigenfunctions remain unchanged once this shift has been applied. The eigenvalue Egj is called the electronic energy. It is known only pointwise for fixed nuclear structures E = but since we may solve Eq. (8.76) for 

Having introduced the electronic wave function we may expand the total state in a product basis as 

The expansion in Eq. (8.77) allows us to solve the eigenvalue equation for the full Hamiltonian H, 

The stationary states Fa) in Sections 2.1 and 2.2 represent general molecular states including all electronic (q) and all nuclear (Q) degrees of freedom. In this section we employ the Bom-Oppenheimer approximation in order to separate the molecular wavefunction into a nuclear part, ,n (Q), and an electronic part, Set(q Q), with the latter depending 

Inserting (2.29) into the time-independent Schrodinger equation (2.5), multiplying with (5%l from the left, and exploiting the orthogonality of the electronic wavefunctions for each nuclear configuration Q readily yields a set of coupled equations for the nuclear wavefunctions (Koppel, Domcke, and Cederbaum 1984), 

Equation (2.31) is still exact. In practice, however, it is extremely difficult to solve. Since Tnu contains first- and second-order derivatives of the 

However, provided the electronic wavefunctions vary only weakly with the nuclear coordinates Q, we may make the approximation 

Within the Born-Oppenheimer approximation the solution of the full problem is thus split into two consecutive problems  

If one or more isotopic substitutions are performed on the moleciile, new frequencies will be obtained, as well as new G matrix elements. Within the Born-Oppenheimer approximation, however, the F matrix elements will transfer intact to the new molecule. The amotint of new information which can be obtained about the elements in the F matrix in this way is, however, limited by several isotope rules rtiich the sets of harmonic frequencies of each symmetry type must obey. One of these, the form of the Teller-Redlich product rule which applies to two isotopic variants having the same molecular symmetry, may be deduced immediately from the secular equation Itself. When the nxn secular determinant is expanded in polynomial form, the constant term, which must be equal to the product of the roots, n.  

In addition to the product rule, there are also sum rules which further restrict the number of independent observables when more than two isotopic variants are available. This subject has been discussed in detail by Heicklen ( ). One interesting conclusion he reaches is that for molecules with symmetry the full F matrix can be determined by substituting for all but one of the sets of equivalent atoms. In principle, then, and 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

In addition to the vibration frequencies and the zeta constants, Aldous and Mills Included in their determination of the force field the data then available on the centrifugal- 

The term ab initio means from first principles it does not mean exact or true . In ab initio molecular orbital theory, we develop a series of well-defined approximations that allow an approximate solution to the Schrodinger equation. We calculate a total wavefunc-tion and individual molecular orbitals and their respective energies, without any empirical parameters. Below, we outline the necessary approximations and some of the elements and principles of quantum mechanics that we must use in our calculations, and then provide a summary of the entire process. Along with defining an important computational protocol, this approach will allow us to develop certain concepts that will be useful in later chapters, such as spin and the Born-Oppenheimer approximation. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H20)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunc-tions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. 

Let us denote by R the coordinates of all the nuclei involved, those of the central ion, its ligands, and the surrounding solvation sphere, and by r the coordinates of all electrons. The Hamiltonian for the 

The product I/(r, R) — R)x(R) is an approximate solution of the Schrodinger equation given by the Hamiltonian H of Eq. (19.1) if terms of order m d j)/dR are neglected. This is justified because the nuclear masses rriN are much larger than the electronic mass me. 

On the other hand, nuclear dynamics on a given potential energy sm-face (PES) V/(R) within the BO separation scheme is reduced to solving the following equation 

Mj and Z/ denote the mass and atomic number of nucleus / nie and e are the electronic mass and elementary charge, and eo is the permittivity of vacuum. The nabla operators Vj and V act on the coordinates ofnudeus I and electron i, respectively. The total wavefunction d (r,I t) simultaneously describes the motion of both electrons and nuclei. 

The Bom-Oppenheimer approximation (Doltsinis and Marx 2002b Kotos 1970 Kutzel-nigg 1997) separates nuclear and electronic motion based on the assumption that the much faster electrons adjust their positions instantaneously to the comparatively slow changes in 

Molecular Dynamics Simulation From Ab Initio to Coarse Grained  

In the classical limit (Doltsinis and Marx 2002b), the nudear wave equation (O 710) is replaced by Newton s equation of motion 

For a great number of physical situations, the Born-Oppenheimer approximation can be safely applied. On the other hand, there are many important chemical phenomena such as charge transfer and photoisomerization reactions, whose very existence is due to the inseparability of electronic and nuclear motion. Inclusion of nonadiabatic effects is beyond the scope of this chapter and the reader is referred to the literature (e.g., Doltsinis 2006 Doltsinis and Marx 2002b) for more details. 

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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... 

Hagedorn, G. A. A time dependent Born-Oppenheimer approximation. Comm. Math. Phys. 77 (1980) 1-19... 

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... 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

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,... 

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