Time-Independent Response Theory

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Perturbative estimate of ESVs with respect to noncorrelated bare Hamiltonian. The specificity of each bond and molecule in the approach based on the SLG expressions for the wave function is taken into account perturbatively by using the linear response approximation [25]. We need perturbative estimates of the expectation values of the pseudospin operators which, in their turn, give values of the density matrix elements according to eq. (3.5). According to the general theory (Section 1.3.3.2) the linear response 5(A) of an expectation value of the operator A to the time independent perturbation AB of the Hamiltonian (A is the parameter characterizing the intensity of the perturbation) has the form ... 

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 time-dependent linear response principle can be applied to the CC theory [22-24], We begin with the Hamiltonian operator H that is a sum of the usual time-independent one 7/"1 and a time-dependent perturbation g(1) ... 

These two categories of methods are essentially identical, differing only in the way some transition properties are defined. Energies given by the two approaches are the same and are obtained by diagonalization of a matrix that can be derived from the EOM point of view, or time-dependent and time-independent response theory. 

In response theory one considers a quantum mechanical system described by the time-independent Hamiltonian which is perturbed by a time-dependent perturbation F(t, e)... 

Response theory is perturbation theory with emphasis on properties rather than on states and energies, aiming to describe how the properties respond to internal or external perturbations. Consider the expectation value of a time-independent operator A, which for the unperturbed system is given by <0 A 0> In the presence of a time-dependent perturbation V(t), the expectation value of A becomes time-dependent and may furthermore be expanded in powers of the perturbation ... 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

The reaction of a many-electron system to a time-dependent perturbation is almost as important as its adjustment to a static time-independent environment change. The theory of the first-order response of a system to an oscillating field follows very similar lines to the perturbation theory of the last chapter. 

In order to complete the set of equations describing polymerization reactions in dense and concentrated regimes, the rates kr and kd must be specified. In the stationary regime, in which the environmental responses to the microscopic motion of the polymer reactants can be assumed to follow the same regression throughout the reaction, these rates are time-independent. The theory of bimolecular reactions in liquids can then be applied at every reaction step. 

These states are formed inside the continuous spectra of the total Hamiltonian and are responsible for phenomena such as resonances in electron scattering from atoms or molecules, autoionization, predissociation, etc. Furthermore, in this work we also consider as unstable states those states that are constructs of the time-independent theory of the interaction of an atom (molecule) with an external field which is either static or periodic, in which case the effect of the interaction is obtained as an average over a cycle. In this framework, the "atom - - field" state is inside the continuous (ionization or dissociation) spectrum, and so certain features of the problem resemble those of the unstable states of the field-free Hamiltonian. The probability of decay of these field-induced unstable states corresponds either to tunneling or to ionization-dissociation by absorption of one or more photons. 

Since time-independent KS theory is commonly used in calculations of molecular structures and properties, we will focus on this form of the theory. We will briefly return to the time-dependent formalism later. The KS Lagrange multipliers in Eqs. [82] and [83] reflect the response of the total electronic energy to changes in occupation number (n/f), i.e.,... 

In response theory, the basis functions %v(f) e usually chosen to be time independent for strong fields or coupled electron-nuclear dynamics, time-dependent basis functions can sometimes be more appropriate. 

Indeed, carbon black-filled rubber, when loaded with time-dependent external forces, suffers a state of stress which is the superposition of two different aspects a time independent, long-term, behavior (sometimes improperly called hyperelastic ) opposed to a time dependent, short-term, response. Step-strain relaxation tests suggest that short term stresses are larger than the long term or quasi-static ones [117]. Moreover, oscillatoiy (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a constitutive relation which depends not only on the current value of the strain but on the entire strain history. This assumption must be in accordance with some principles which restrict the class of rehable constitutive equations. These restrictions can be classified as physical and constitutive . The former are restrictirMis to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration, e.g., internal symmetries. 

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