Perturbation theory

The use of MPT in ionic solution theory will ensure good low-p results if one simply takes the union of our F-ordered result with the lowest-order p-ordered result, which is described in detail by Friedman and by Kelbg. That is, one should add to our result the part of the lowest-ordered p-ordered result that is not already contained in our result. This means adding to our SOGA result for the free energy the term in S 

The function (12, p) is analytically trivial. It is simply the Debye-Huckel correlation function 

Here is the charge on the particle at r which is of species i and p, is the concentration of that species. The corresponding term to be added to either of our simplified SOGA results for g(l 2) [called QUAD and in the 

As soon as p becomes appreciable (p/ ) (95) and (98) become negligibly small. This corresponds to concentrations above about molar for 1-1 electrolytes. 

The quantity P is a constant for a given series of polymer homologs. Equations (5.19a) and (5.19b) are Flory s theory to describe the polymer chain configuration, de Gennes commented that because of the assumption of Gaussian distribution with respect to R )q, Flory s theory is still intrinsically a random-flight chain in nature. 

Perturbation theory (Zimm et al., 1953 Yamakawa, 1971) is an extension of Hory s theory. It is based on mathematical analysis of the transition probabihty, which indicates that the excluded volume effect on R ) becomes asymptotically proportional to a power of N higher than the first. Flory s term a may be expanded in a power series by the excluded volume parameter z- The underlying idea is to describe a as a universal function of z. For this reason, the theory is also called the two-parameter (a and z) theory. So far it has only partially reached this goal, because it is a many-body problem and the mathematics involved are difficult 

We now discuss the second major quantum-mechanical approximation method, perturbation theory. 

Suppose we have a system with a time-independent Hamiltonian H and we are unable to solve the Schrbdinger equation 

We shall call the system with Hamiltonian the unpertuilted system. Hie system with Hamiltonian H is the perturbed system. The difference between the two Hamiltonians is the perturbation, ft  

When A is zero, we have the unperturbed system. As A increases, the perturbation grows larger, and at A = 1 the perturbation is fully turned on. We have introduced A as a convenience in relating the perturbed and unperturbed eigenfunctions, and ultimately we shall set A = 1, thereby eliminating it. 

Sections 9.1 to 9.8 deal with time-independent Hamiltonians and stationary states. Section 9.9 deals with time-dependent perturbations. 

An alternative approach toward including electron correlation is provided by perturbation theory. Suppose we have an operator 0 that can be decomposed into two component operators 

M0ller and Plesset developed the means for applying perturbation theory to molecular system. They divided the full Hamiltonian (Eq. (1.5)) into essentially 

Nonetheless, MP2 is quite a bit slower than HF theory. The resolution of the identity approximation (RI) makes MP2 nearly competitive with HF in terms of computational time. This approximation involves a simplification of the evaluation 

Grimme proposed an empirical variant of MP2 that generally provides improved energies. This is the spin-component-scaled MP2 (SCS-MP2) that scales the terms involving the electron pairs having the same spin (SS) differently than those with opposite spins (OS). The SCS-MP2 correlation correction is given as 

Let us first follow London s original procedure, that is, let us solve the two particle Schrodinger equation by means of perturbation theory. We consider two particles 1 and 2 at positions and j-j, which exhibit the one-electron Hartree-Fock or pseudopotentials I/i(r) and U2(r), as shown in Fig. 27. We denote the one-electron orbitals localized at particle 1 by i , /c , m . those localized at particle 2 by [/ , / , 

We are interested in studying the correlation energy of the two-electron orbitals of particles 1 and 2. Thus, turning to finite separations 21 = 2 — i we consider the Schrodinger equation 

The interaction potential W(ri,r) is built up from the exchange terms C/i(o) and U2(i, ) such that electron i sees particle 2 and electron j sees particle 1, and from the Coulomb repulsion of the electrons F,(rj, rj) plus that of the remaining ions rj), i.e. 

In the following we assume the separation r2i to be large enough to exclude overlap of electron orbitak localized at different particles. In this case it is permissible to omit all exchange terms in the two-electron orbitals v (i-j, r ) and to write 

The two-electron orbital v(i (,i y) is a linear combination of products of one-electron orbitals localized at different particles. Applying second order perturbation theory to the Schrodinger equation (7.4), the energy of the orbital specified by (7.6) is found to be 

In the original formula of Flory, a = 2.6(3/27t). Stockmayer suggested that the choice of a = 4/3(3/2ti) gives the exact result of first order perturbation theory near 6 temperature and the result of equation (23) in good solutions. With this choice of a, equation (24) is called the modified Flory formula which can be derived in a systematic way. Many approximate closed expressions for a exist in the literature which are adequately reviewed.  

The full theory of this is complex and controversial. We will give a brief account. Introduction of three-body interactions into the analysis gives (instead of equation 24) 

There is a mathematical formalism ideally suited to describing the global properties of polymers known as path integration. In this formalism, the polymer is described by a position function r(s) of the arc length s which is the continuum limit of the label n of the nth monomer. 

The probability distribution function G R, L) of the end-to-end distance / of a chain with the excluded volume interaction is given by the path integral  

The expression jD[r] means the continuum limit of 7idr . Another way to look at this is to realize that J D[r] implies the summation over all possible paths between the ends of the chain KO) which is taken to be the origin and r(L) = R. 

When the Hartree-Fock wave function provides a reasonably accurate description of the electronic structure, it is tempting to try and improve on it by the application of perturbation theory. Indeed, this approach to the correlation problem has been quite successful in quantum chemistry - in particular, in the form of M0ller-Plesset perturbation theory, to which we now turn our attention. 

In M0ller-Plesset perturbation theory (MPPT), the electronic Hamiltonian H in (5.1.2) is partitioned as 

In MPPT, the Fock operator represents the zero-order operator and the fluctuation potential the perturbation. The zero-order electronic state is represented by the Hartree-Fock state in the canonical representation 

Applying the standard machinery of perturbation theory, we obtain to second order in the perturbation 

The MP2 model represents a highly successful approach to the correlation problem in quantum chemistry, providing a surprisingly accurate, size-extensive correction at low cost. Higher-order corrections may be derived as well. The MP3 and MP4 corrections, in particular, have found widespread use but represent less successful compromises between cost and accuracy than does the MP2 correction. For a detailed exposition of perturbation theory, we refer to Chapter 14. 

Indeed for the Neel-based theory to work best it is better to have a bipartite system (i.e., a system with two sets of sites all of either set having solely only members from the other set as neighbors). Of course, when there is a question about the adequacy of the zero-order description questions about the (practical) convergence of the perturbation series arises. But for favorable systems these [38] or closely related [41] expansions can now be made through high orders to obtain very accurate results. 

One can also imagine a perturbation expansion based on the resonating VB limit as zero-order. This has been considered [23,24] for some general circumstance. But for the covalent space for the n-networks of benzenoids the different Kekule structures would all be degenerate, and the degenerate perturbation theory can be neatly 

The number of problems that can be solved exactly in mechanics is not large. Once we have to treat three interacting bodies, life becomes very difficult indeed. This comment applies to classical mechanics just as to quantum mechanics. What we often do is to look for a simple, idealized problem that we can solve exactly, and then treat the real problem in hand as some kind of perturbation on the idealized one. 

In perturbation theory, we write the true Hamiltonian H in terms of and a perturbation 

I have included the arbitrary parameter X in order to keep track of orders of magnitude. I will later set it to unity. In the case of the helium problem above, the perturbation would be just the Coulomb repulsion between the electrons. 

Very occasionally, we might need to add more than one perturbation foj-example, if we wanted to study a molecule subject to external electric and magnetic fields, we might write something like 

Back now to the simpler case of a single perturbation. Perturbation theory aims to write solutions for 

While the multiconfiguration methods lead to large and accurate descriptions of atomic states, formal insight that can lead to a productive understanding of structure-related reaction problems can be obtained from first-order perturbation theory. We consider the atomic states as perturbed frozen-orbital Hartree—Fock states. It is shown in chapter 11 on electron momentum spectroscopy that the perturbation is quite small, so it is sensible to consider the first order. Here the term Hartree—Fock is used to describe the procedure for obtaining the unperturbed determi-nantal configurations pk). The orbitals may be those obtained from a Hartree—Fock calculation of the ground state. A refinement would be to use natural orbitals. 

The Hartree—Fock Hamiltonian is written as a sum over electrons with coordinate—spin labels s. 

The Schrodinger equation for the atomic state i) is written in the form 

If the perturbation V is reduced to zero then Ei is the same as the Hartree—Fock energy Ei. This is used to define the label i for a Hartree-Fock configuration pi). It is the configuration of the same symmetry as i) whose energy Ei is nearest to The residual electron—electron potential V splits the Hartree—Fock energy levels so that there is more than one atomic state for every Hartree—Fock state. 

We multiply the Schrodinger equation (5.75) on the left by the inverse of the Hartree—Fock operator to obtain 

Gravity is a force of infinite range, and it is impossible for any pair of objects to be truly isolated and subject to a point mass central field. The closed form solution of the two-body problem thus represents an idealized orbit. The departures from this trajectory are treated by perturbation theory. The action of any additional mass in a system can 

Perturbations introduced by the action of external bodies fall into several categories. For examle, the effect of finite size of the central object in a two-body problem introduces precession in an orbit that can be treated as a perturbation above the point central field. These orbital perturbations represent simple time-independent and periodic departures from the closed ellipse. They will cause the orbiting body to evolve toward a stable trajectory if the central body is not rotating. Rotation of the central body introduces an additional time scale into the problem and can produce secular instabilities in the orbit. The basic starting point of a perturbation calculation is that one already knows what the orbit is for a particle. One is interested in finding out whether it is stable against small perturbations, due to other bodies, and what the evolution will be for the orbit. 

Tidal perturbations are important for the orbits of many satellites. In the course of time, a satellite gains angular momentum through interaction with the sun and moon, as well as because of the nonspherical gravitational field of the earth. Orbital ephemerides must be frequently updated to take these changes into account, especially for geosynchronous satellites. 

The interaction of the sun and moon with the earth is responsible for several important physical effects, notably the precession of the rotation axis with time (the phenomenon first described by Hipparchos) and for the change in the length of the day due to tidal friction and dissipation of rotational energy. The tidal term results from the finite size of the earth relative to its orbital radius and to that of the moon. The differential gravitational acceleration across the body produces a torque that accelerates the moon outward and slows the earth s rotation. 

Notice that the variation of the semimajor axis depends on both the radial and the torque, but because the orbit can be taken in the two-body problem as planar, there is no dependence on W. Changes in co and are equivalent to orbital precession. All of these may be periodic or secular, depending on the details of 91. For a given disturbing function, this system of equations can be well explored using numerical methods. 

In this chapter we return to the problem of constructing models for covalent bonding and develop further some of the ideas reviewed in Chapter 1 in order to prepare for the treatment of pericyclic reactions in Chapter 11. 

In Chapter 1 we considered only interactions among atomic orbitals. It would be useful, however, to be able also to deal with interactions between molecular orbitals. For example, suppose that we have two molecules of ethylene, which are approaching each other as shown in 1, so that the n orbitals of the two molecules come in closer and closer contact. We might ask whether the following reaction will occur  

In order to answer this question, we need to be able to tell how the orbital energies of each molecule will be altered by the presence nearby of the orbitals of the other molecule. The situation is thus just the same as in bringing together two hydrogen atoms, except that here we are talking about molecular orbitals instead of atomic orbitals. 

Fortunately, the same rules apply to interaction between two molecular orbitals as apply to interaction between two atomic orbitals. We shall be able to make a new model to cover the interacting situation by adding and subtracting the molecular orbital functions that were correct for the separate molecules before the interaction occurred. 

From here on, then, everything we say will apply to orbitals in general, be they atomic or molecular. 

Amos and G. G. Hall, Proc. Roy. Soc., 1961, A263, 483 A. T. Amos and L. C. Snyder, J. Chem. Phys., 1964, 41, 1773. There is a trivial error in some of the formulae in the latter reference, a serious error in some of the formulae in the former  

Smeyers and G. Delgado-Barrio, Internal. J. Quantum Chem., 1974, 8, 733. 

Graphed as a function s this should be linear with slope —/3. The intercept then provides the thermodynamic parameter sought. 

Similarly, if chemical potential contributions in excess of a defined reference system are sought, then 

To organize the description of interactions of a specified type, it is often helpful to introduce an ordering parameter A in 

A might also be viewed as a perturbative parameter in cases where it appears naturally as a gauge of the strength of solute-solvent interactions. In either case. 

This expansion in powers of A can be viewed from a more general perspective. The counting and arranging of powers of A is a formal operation, so we can carry out that analysis on the basis of a simpler notation, e.g. 

FIGURE 12.6 A convenient way to remember the order of filling of the subshells in most atoms (1-85). Simply follow the order of subshells crossed by the arrows. 

In a previous section, we presumed that the wavefunctions of multielectron atoms can be approximated as products of hydrogen-like orbitals  

Unless otheiwise noted, all art on this page is Cengage Learning 2014. 

TABLE 12.1 Ground-state electron configurations of the elements  

In the previous chapter we have derived the molecular electronic Hamiltonian H in the presence of static electromagnetic fields or fields due to nuclear moments. Throughout this chapter we will consider a general field and denote it as T with components J-a - -. Examples for. Fq. are one of the three components of the electric field a, of the magnetic induction Ba, of the nuclear moment of a magnetic nucleus K or one of the nine components of the field gradient Sa/3-  

Our first task in this chapter is to obtain expressions for the wavefunction 4 o(.F)) of the ground state of our system in the presence of the components of such a static field and afterwards expressions for the energy Eq T) and for the expectation value ( o(F) IPI o(F)) of an arbitrary operator P in the presence of the field. This means that we have to solve the time-independent Schrodinger equation for the system 

Our second task is to generalize this approach to the case of time-dependent fields and the solution of the time-dependent Schrodinger equation 

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The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

In perturbation theory, the Hamiltonian is divided into two parts. One of these eorresponds to a Selirodinger equation that ean be solved exaetly... 

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

In the quantum mechanics of atoms and molecules, both perturbation theory and the variational principle are widely used. For some problems, one of the two classes of approach is clearly best suited to the task, and is thus an established choice. Flowever, in many others, the situation is less clear cut, and calculations can be done with either of the methods or a combination of both. 

There has been a great deal of work [62, 63] investigating how one can use perturbation theory to obtain an effective Hamiltonian like tlie spectroscopic Hamiltonian, starting from a given PES. It is found that one can readily obtain an effective Hamiltonian in temis of nomial mode quantum numbers and coupling. 

Perturbation theory is a natural tool for the description of intemioleciilar forces because they are relatively weak. If the interactmg molecules (A and B) are far enough apart, then the theory becomes relatively simple because tlie overlap between the wavefiinctions of the two molecules can be neglected. This is called the polarization approximation. Such a theory was first fomuilated by London [3, 4], and then refomuilated by several others [5, 6 and 7]. 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

Perturbation theory yields a siim-over-states fomnila for each of the dispersion coefficients. For example, the isotropic coefficient for the interaction between molecules A and B is given by... 

In the third order of long-range perturbation theory for a system of tluee atoms A, B and C, the leading nonadditive dispersion temi is the Axilrod-Teller-Mutd triple-dipole interaction [58, 59]... 

The perturbation theory described in section Al.5.2,1 fails completely at short range. One reason for the failure is that the multipole expansion breaks down, but this is not a fiindamental limitation because it is feasible to construct a non-expanded , long-range, perturbation theory which does not use the multipole expansion [6], A more profound reason for the failure is that the polarization approximation of zero overlap is no longer valid at short range. 

Claverie [M] has argued that if perturbation theory converges at all, it will converge to one of these... 

A fiirther diflfieulty arises beeause the exaet wavefiinetions of the isolated moleeules are not known, exeept for one-eleetron systems. A eoimnon starting point is the Hartree-Foek wavefiinetions of the individual moleeules. It is then neeessary to inelude the eflfeets of intramoleeular eleetron eorrelation by eonsidering them as additional perturbations. Jeziorski and eoworkers [M] have developed and eomputationally implemented a triple perturbation theory of the syimnetry-adapted type. They have applied their method, dubbed SAPT, to many interaetions with more sueeess than might have been expeeted given the fiindamental doubts raised about the method. SAPT is eurrently both usefiil and praetieal. A reeent applieation [ ] to the CO2 dimer is illustrative of what ean be aehieved widi SAPT, and a rieh soiiree of referenees to previous SAPT work. 

Wormer P E S and Hettema H 1992 Many-body perturbation theory of frequency-dependent... 

Jeziorski B, Moszynski R and Szalewicz K 1994 Perturbation theory approach to intermolecular potential energy surfaces of van der Waals complexes Chem. Rev. 94 1887... 

Adams W H 1994 The polarization approximation and the Amos-Musher intermolecular perturbation theories compared to infinite order at finite separation Chem. Phys. Lett. 229 472... 

Bukowski R, Sadie] J, Jeziorski B, Jankowski P, Szalewicz K, Kucharski S A, Williams H L and Rice B M 1999 Intermolecular potential of carbon dioxide dimer from symmetry-adapted perturbation theory J. Chem. Phys. 110 3785... 

Hayes I C and Stone A J 1984 An intermolecular perturbation theory for the region of moderate overlap Mol. Phys. 53 83... 

Stone A J 1993 Computation of charge-transfer energies by perturbation theory Chem. Phys. Lett. 211 101... 

If //j is small compared with EI we may treat EI by perturbation theory. The first-order perturbation theory fomuila takes the fonn [18, 19, 20 and 21] ... 

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

In perturbation theories of fluids, the pair total potential is divided into a reference part and a perturbation... 

A very successfiil first-order perturbation theory is due to Weeks, Chandler and Andersen pair potential u r) is divided into a reference part u r) and a perturbation w r)... 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

Perturbation theory is also used to calculate free energy differences between distinct systems by computer simulation. This computational alchemy is accomplished by the use of a switching parameter X, ranging from zero to one, that transfonns tire Hamiltonian of one system to the other. The linear relation... 

It follows from our previous discussion of perturbation theory that... 

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