R-adiabaticity

Mastrangelo, S.V.R. Adiabatic calorimeter for determination of cryoscopic data. Anal. Chem., 29(5) 841-845, 1957. 

Fig. 2.2 A schematic description of the LZ problem Two quanmm states and the coupling between them depend parametrically on a classical variable ff. The energies of the zero-order states a and b cross at R = R. The energies obtained by diagonalizing the Hamiltonian at any point R (adiabatic states) are ] (R) aDdE2(R).

The action over the double well is quantized by a (2n+l)h action condition. In other words, the resonant orbit with (2n+l)h action will give the adiabatic well energy and location of the U-(r) adiabatic potential energy curve. For example, in Fig. 6 we compare the energies and locations of quantized RPO s of the IHl system with the quantal (averaged) adiabatic surfaces. 

Table 39.2 Excitation levels, equilibrium intemuclear distances (r ), adiabatic excitation energies (T ) and main configurations for the singlet and triplet states of C2...

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

Sircar, S.. and Kumar. R., Adiabatic adsorption of bulk binary gas mixtures Analysis by constant pattern model. Ind. Eng, Chem. Process Des, Dev.. 22(2), 271-280 (1983). 

Hofmann A, de Vivie-Riedle R. Adiabatic approach for ultrafast quantum dynamics mediated by simultaneously active conical intersections. Chem Phys Lett. 2001 346 299-304. 

At this point, we restrict the multi-state quantum chromophore to the ground ( 0 R)), first-excited ( 1 R)), and second-exdted ( 2 R)) adiabatic states, which is the minimum number of states required to describe third-order optical response in ultrafast IR experiments. It is assumed that only transitions between 0 R) and 11 . R) and between 11 i ), and 2 R) are allowed. Also, the rotating wave approximation (RWA) is invoked, such that the field-matter interaction term may be written as ... 

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

Quack M and Troe J 1998 Statisticai adiabatic channei modeis Enoyolopedia of Computational Chemistry ed P v R Schieyer et a/(New York Wiiey) pp 2708-26... 

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

A calorimeter is a device used to measure the work that would have to be done under adiabatic conditions to bring about a change from state 1 to state 2 for which we wish to measure AU= U -U This work is generally done by passing a known constant electric current 3 for a known time t through a known resistance R embedded in the calorimeter, and is denoted by where... 

Magee J W, Blanco J C and Deal R J 1998 High-temperature adiabatic calorimeter for constant-volume heat capacity of compressed gases and liquids J. Res. Natl Inst. Stand. Technol. 103 63... 

Solution of this set for F R) represents tire adiabatic close-coupling method. The adiabatic states are nomrally detennined (via standard computational teclmiques of quanUim chemistry) relative to a set of axes (X, Y, Z ) with the Z- axis directed along the nuclear separation R. On transfomring to this set which rotates during the collision, then /(r, / ), for the diatomic A-B case, satisfies... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

Quack M and Tree J 1998 Statistical adiabatic channel models Encyciopedia of Computationai Chemistry vol 4, ed P v R Schleyer et ai (New York Wiley) pp 2708-26... 

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... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

Baer R, Charutz D M, Kosloff R and Baer M 1996 A study of conical intersection effects on scattering processes—the validity of adiabatic single-surface approximations within a quasi-Jahn-Teller model J. Chem. Phys. 105 9141... 

Niv M Y, Krylov A I and Gerber R B 1997 Photodissociation, electronic relaxation and recombination of HCI in Ar-n(HCI) clusters—non-adiabatic molecular dynamics simulations Faraday Discuss. Chem. Soc. 108 243-54... 

Wynne K and Hochstrasser R M 1999 Coherence and adiabaticity in ultrafast electron transfer Adv. Chem. Phys. 107 (Electron transfer from isolated molecules to biomolecules) part 2, 263-309... 

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. 

V is the derivative with respect to R.) We stress that in this formalism, I and J denote the complete adiabatic electronic state, and not a component thereof. Both /) and y) contain the nuclear coordinates, designated by R, as parameters. The above line integral was used and elaborated in calculations of nuclear dynamics on potential surfaces by several authors [273,283,288-301]. (For an extended discussion of this and related matters the reviews of Sidis [48] and Pacher et al. [49] are especially infonnative.)... 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

Requiring l/f (r qx) to be real, the matrix W (Rx) becomes real and skew-symmetiic (just like its adiabatic counterpart) with diagonal elements equal to zero. Similarly, W (Rx) is an n X u diabatic second-derivative coupling matrix with elements defined by... 

It needs mentioning that the diabatic Schrodinger equation (31) also contains a gradient term W (Rx) Vr x(R ) like its adiabatic counterpart [Eq. (15)]. 

---