Adiabatic states infinite number

As written, Eq. (52) depends on all the (infinite number of) adiabatic electi onic states. Fortunately, the inverse dependence of the coupling strength on energy separation means that it is possible to separate the complete set of states into manifolds that effeetively do not interact with one another. In particular, Baer has recendy shown [54] that Eq. (57), and hence Eq. (58) also holds in the subset of mutually coupled states. This finding has important consequences for the use of diabatic states explored below. 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

Figure 4.13 Adiabatic hyperspherical potentials, Eq. (101), in a.u. without the adiabatic correction term for He of symmetries 1,3SC and 1,3P° converging to the asymptotic limit e + He+(n = 2), plotted against the hyperradius R in a.u. Each potential supports an infinite number of Rydberg states of Feshbach resonance, of which the lowest level is indicated by a horizontal line. Figure from Ref. [90], Note the difference in notation.

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

The existence of states that are inaccessible to adiabatic processes was shown by Carath odory to be necessary and sufficient for the existence of an integrating factor that converts into an exact differential [2-4]. From the calculus we know that for differential equations in two independent variables, an integrating factor always exists in fact, an infinite number of integrating factors exist. Experimentally, we find that for pure one-phase substances, only two independent intensive properties are needed to identify a thermod)mamic state. So for the experimental situation we have described, we can write SQ gj, as a function of two variables and choose the integrating factor. The simplest choice is to identify the integrating factor as the positive absolute thermodynamic temperature X = T. Then (2.3.3) becomes... 

Now consider the initial and final states of a reversible process with one-way heat (i.e., each nonzero infinitesimal quantity of heat has the same sign). Since we have seen that it is impossible for there to be a reversible adiabatic path between these states, the points for these states must lie on different reversible adiabatic surfaces that do not intersect anywhere in the A-dimensional space. Consequently, there is an infinite number of nonintersecting reversible adiabatic surfaces filling the A-dimensional space. (To visualize this for A = 3, think of a flexed stack of printer paper each sheet represents a different adiabatic surface in three-dimensional space.) A reversible, nonadiabatic process with one-way heat is represented by a path beginning at a point on one reversible adiabatic surface and ending at a point on a different surface. If q is positive, the final surface lies on one side of the initial surface, and if q is negative, the final surface is on the opposite side. 

The curl condition given by Eq. (43) is in general not satisfied by the n x n matrix W (R ), if n does not span the full infinite basis set of adiabatic elechonic states and is huncated to include only a finite small number of these states. This tmncation is extremely convenient from a physical as well as computational point of view. In this case, since Eq. (42) does not have a solution, let us consider instead the equation obtained from it by replacing WC) (R t) by its longitudinal part... 

Equations of state (EoS) are an attempt to capture the interrelationships between the variable set of thermodynamics for infinite matter, in the absence of long-range interactions. Thermodynamics itself is the enterprise of trying to describe matter with the absolute minimum number of macroscopic variables. One deduces the EoS from direct measurement of the macroscopic variables and from measureable derivatives such as the isothermal and adiabatic compressibihties for the mechanical sector and the constant pressure or volume heat capacities for the thermal sector. 

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