Time-Independent Perturbation Theory

Perturbation theory builds on the partitioning of the Hamiltonian from Eq. (2.101) in an unperturbed Hamiltonian and perturbation Hamiltonians + X H  

We suppose now that the energies and wavefunctions (n = 0,1,2, ) of the unperturbed Hamiltonian are known, i.e. that the unperturbed Schrodinger equation 

Finally, we assmne that the eigenfunction o(. )) nd eigenvalue Eo iF) of the full Hamiltonian H are close to those of the unperturbed Hamiltonian i.e. the perturbation by T is indeed small. We can then expand the perturbed wavefunction 

It is called the generalized Hellmann-Feynman theorem because Hellmann (1937) and Feynman (1939) considered originally the changes in energy due to a change in the geometry. 

The energy e E) and wavefunction o (.F)) are called the mth-order correction to the energy and wavefunction. Terms like or are 

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Suggested Extra Reading- Appendix D Time Independent Perturbation Theory]... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

Just as with time-independent perturbation theory, we can go to higher-order approximations if necessary. See Fong, pp. 234-244. 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

M0LLER-PLESSET (MP) TIME-INDEPENDENT PERTURBATION THEORY... 

Taking these introductory comments as a motivation, we shall turn to the formalism of response theory. Response theory is first of all a way of formulating time-dependent perturbation theory. In fact, time-dependent and time-independent perturbation theory are treated on equal footing, the latter being a special case of the former. As the name implies, response functions describe how a property of a system responds to an external perturbation. If initially, we have a system in the state 0) (the reference state), as a weak perturbation V(t) is turned on, the average value of an operator A will develop in time according to... 

The expressions derived for the molecular properties in the previous section are of a rather general and perhaps somewhat abstract character. For a given variational wave function, the explicit expressions for the molecular properties are obtained by substituting in Eqs. 18 to 21 the detailed form of the energy functional (x A) for a nonvariational wave function, we first express the energy as a variational Lagrangian and then proceed in the same manner. We shall not discuss the detailed expressions for the derivatives here, referring instead to special reviews [1]. Still, to illustrate the physical contents of Eq. 18 and Eq. 21, we shall now see how these expressions are related to those of standard time-independent perturbation theory. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

One of the most useful techniques in physics and chemistry is Time Independent Perturbation Theory (TP). Therefore, not surprisingly, it has been applied to the study of the CHA problem. 

The exfernal magnetic field Mq can be treated as a small perturbation by time-independent perturbation theory. 

Use of the approximate result (9.121) for the expansion coefficients in (9.118) gives the desired approximation to the state function at time t for the case that the time-dependent perturbation H is applied at r = 0 to a system in stationary state n. [As with time-independent perturbation theory, one can go to higher-order approximations (see Fong, pp. 234-244).]... 

The form of Jf, given by Eq. (3), is not very helpful with regard to understanding the relationship of Jf to H. There are many different ways in which to formulate this problem, in terms of time-independent perturbation theory [8,9,13-15], time-dependent perturbation theory [8,9,16,17], the coupled-cluster or e method [18], moment methods [19], and variational approaches [20]. Because of time and space restrictions, we will discuss in detail only the time-independent perturbation-theory approach. Those interested in other techniques should peruse the appropriate references. 

The perturbed final state wave function in Fe (aq) ion was expressed using the time-independent perturbation theory as ... 

We assume the reader has a good background in undergraduate quantum mechanics and electrostatics. In particular, the ideas of matrix mechanics, Dirac notation, and time-independent perturbation theory are used fi-equently. In addition, the reader should have a passing familiarity with electric dipoles and their interactions with fields and with each other. Finally, we will draw heavily on the mathematical theory of angular momentum as applied to quantum mechanics in more detail in the classic treatise of Brink and Satchler [1]. When necessary, details of the structure of diatomic molecules have been drawn from Brown and Carrington s recent authoritative text [2]. 

Connection Between Time-Independent Perturbation Theory and Spectroscopic Selection Rules... 

We recover the dispersion energy (7.22) obtained by applying time-independent perturbation theory to the Schrodinger equation, and find an additional factor coth(fioi/2fcr) accounting for the intensity of the interacting fields. 

Although time-independent perturbation theory is mostly used for the derivation of energy corrections, it is not restricted to this but can be applied to any physical observable P and its expectation value as defined in Eq. (2.8). In this section, we want to illustrate this for the case of an observable whose corresponding operator is obtained as a derivative of the Hamiltonian with respect to the component Pa - of a field P, i.e. the field-dependent operator P ) defined in Eqs. (3.8) to (3.10). 

The last three terms are linear in the components of the field and represent the linear response of the expectation value P Tg° ) to the field f. Therefore, we want to call this application of time-independent perturbation theory to the case of an expectation value in the presence of a field also time-independent response theory. 

The compact integral expressions for the time-dependent wavefunction in Eq. (3.86) and in particular for the first-order correction in Eq. (3.87) will be employed in the derivation of response functions in the following section. However, for the interpretation of the time-dependent wavefunctions it is useful to expand them in the complete set of unperturbed wavefunctions Eq. (2.14) analogous to the perturbed wavefunctions of time-independent perturbation theory in Eq. (3.23)... 

Comparison with Eq. (3.33) shows that the second-order correction in pseudoperturbation theory has essentially the same structure as in regular time-independent perturbation theory. 

In the first part of the book we have derived the Hamiltonian for the interaction of molecules with electromagnetic fields. Furthermore, we have employed time-independent perturbation theory or static response theory in order to obtain expressions for the corrections to the energy and wavefunction of a molecule due to the interaction with electromagnetic fields. We are thus well prepared for defining many different molecular properties in this and the following chapters and for deriving quantum mechanical expressions for them. 

Using time-independent perturbation theory from Section 3.2 or response theory as described in Sections 3.3 and 3.11 one can derive the following expressions for the first-order... 

By using the time-independent perturbation theory and the HF method as starting point, Moller and Plesset obtained, to seeond order (MP2), an estimation to the correlation energy, as... 

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