Many body response theory

Note that the many-body response functions are in general non-local, implying that the response (i.e. the polarization) at point r depends on the electric field at other locations. This makes sense In a system of interacting molecules the response of molecules at location r arises not only from the field at that location, but also from molecules located elsewhere that were polarized by the field, then affected other molecules by their mutual interactions. Also note that by not stressing the vector forms of and P we have sacrificed notational rigor for relative notational simplicity. In reality the response function is a tensor whose components are derived from the components of the polarization vector, and the tensor product ... / is the corresponding sum over vector components of and tensor components of /. 

---

Trucks GW, Salter EA, Noga J, Bartlett RJ (1988) Analytic many body perturbation theory MBPT(4) response properties. Chem Phys Lett 150 37 14... 

Linear-response coupled-cluster Many-body perturbation theory of order n Molecular orbital... 

The quantity x is the linear density-density response function of the system. In other branches of physics it has other names, e.g., in the context of many-body perturbation theory it is called the reducible polarization function. Unfortunately, the evaluation of x through perturbation theory is a very demanding task. We can, however, make use of TDDFT to simplify this process. 

Quantum Systems in Chemistry and Physics is a broad area of science in which scientists of different extractions and aims jointly place special emphasis on quantum theory. Several topics were presented in the sessions of the symposia, namely 1 Density matrices and density functionals 2 Electron correlation effects (many-body methods and configuration interactions) 3 Relativistic formulations 4 Valence theory (chemical bonds and bond breaking) 5 Nuclear motion (vibronic effects and flexible molecules) 6 Response theory (properties and spectra atoms and molecules in strong electric and magnetic fields) 7 Condensed matter (crystals, clusters, surfaces and interfaces) 8 Reactive collisions and chemical reactions, and 9 Computational chemistry and physics. 

Figure 6.14 compares the results of line shape computations based on the isotropic interaction approximation with the measurement by Hunt [187], This spectrum does not have many striking features because of the relatively high temperature of 300 K. We notice only a broad, unresolved Q branch and a diffuse Si(l) line of H2 is seen other lines such as Si(J) with J = 0, 2, 3,. .. are barely discernible. Various dips of the absorption at 4126, 4154 and 4712 cm-1 are caused by intercollisional interference, a many-body effect which is not accounted for in a binary theory. Roughly 90% of the Q branch (in the broad vicinity of 4150 cm-1) arises from the isotropic overlap induced dipole component (XL = 01). The anisotropic overlap component (XL = 21) is a little less than one-half as intense as the quadrupole induced term (XL = 23). These two components with X = 2 are responsible for the Si line structures superimposed on the broad isotropic induction component which is of roughly comparable intensity near the Si line center. 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

The time-dependent density functional theory, widely known as TDDFT, is an exact many-body theory [1] in which the ground state time-dependent electron density is the fundamental variable. For small changes in the time-dependent electron density, a linear response (LR) approach can be applied to solve the TDDFT equations. In... 

Absorption of the X-ray makes two particles in the solid the hole in the core level and the extra electron in the conduction band. After they are created, the hole and the electron can interact with each other, which is an exciton process. Many-body corrections to the one-electron picture, including relaxation of the valence electrons in response to the core-hole and excited-electron-core-hole interaction, alter the one-electron picture and play a role in some parts of the absorption spectrum. Mahan (179-181) has predicted enhanced absorption to occur over and above that of the one-electron theory near an edge on the basis of core-hole-electron interaction. Contributions of many-body effects are usually invoked in case unambiguous discrepancies between experiment and the one-electron model theory cannot be explained otherwise. Final-state effects may considerably alter the position and strength of features associated with the band structure. 

The earliest quantitative theory to describe van der Waals forces between two colloidal particles, each containing a statistically large number of atoms, was developed by Hamaker, who used pairwise summation of the atom-atom interactions. This approach neglects the multi-body interactions inherent in the interaction of condensed phases. The modem theory for predicting van der Waals forces for continua was developed by Lifshitz who used quantum electrodynamics [19,20] to account for the many-body molecular interactions and retardation within and between materials. Retardation is a reduction of the interaction because of a phase lag in the induced dipole response that increases with distance. 

The CT/ET free energy surface is the central concept in the theory of CT/ ET reactions. The surface s main purpose is to reduce the many-body problem of a localized electron in a condensed-phase environment to a few collective reaction coordinates affecting the electronic energy levels. This idea is based on the Born-Oppenheimer (BO) separation " of the electronic and nuclear time scales, which in turn makes the nuclear dynamics responsible for fluctuations of electronic energy levels (Eigure 1). The choice of a particular collective mode is dictated by the problem considered. One reaction coordinate stands out above all others, however, and is the energy gap between the two CT states as probed by optical spectroscopy (i.e., an experimental observable). 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

The other category of study focuses on the nature of the transfer in the condensed phase and in biological systems. Here, it is not perhaps beneficial to consider every atom of a many-body complex system. Instead, the objective is hopefully to project the key electronic and nuclear forces which are responsible for behavior. With this perspective, approximate, but predictive, theories have a much more valuable outreach in applications than those simulating or computing bonding and motion of all atoms. Computer simulations are important, but for such systems they should be a tool of guidance to formulate a predictive theory. Similarly for experiments, the most significant ones are those that dissect complexity and provide lucid pictures of the key and relevant processes. 

---