Finite field

The electric field can be incorporated in the Flamiltonian via a finite field term or approximated by a set of point charges. This allows the computation of corrections to the dipole only, which is generally the most significant contribution. 

In practice the finite-field calculation is not so simple because the higher-order terms in the induced dipole and the interaction energy are not negligible. Normally we use a number of applied fields along each axis, typically multiples of 10 " a.u., and use the standard techniques of numerical analysis to extract the required data. Such calculations are not particularly accurate, because they use numerical methods to find differentials. 

As an example, here is an output from Gaussian 98 on CH3F (Figure 17.2). I forced the finite field method by choice of Polar = Enonly (Polar = Energy only) in the route. The geometry was first optimized and stored in a checkpoint file. 

Most of the analytical structure of the dynamics of linear CA systems emerges from their field-theoretic properties specifically, those of finite fields and polynomials over fields. A brief summary of definitions and a few pertinent theorems will be presented (without proofs) to serve as reference for the presentation in subsequent sections. 

Two simple examples of fields are the rational numbers, Q and the set of integers Zp = (0,1,..., p — 1, where p is prime and addition and multiplication are defined modulo p. The latter is an example of a finite field - that is, of a field containing only a finite number of elements. Since CA are almost always defined so that individual sites take on one of a finite number of values, if those values happen to be elements of a finite field then the dynamics can be well understood by using some of the general properties of that field. An elementary property of finite fields... 

We will have occasion to use both polynomials and matrices defined over some finite field Fg. In the case of a polynomial f x), what is meant is simply that... 

It is easy to show that every polynomial f x) (not divisible by x) over a finite field J g is a factor of 1—cc , for some power n . The order (sometimes also called the period or exponent) of f x), denoted by ord(/), is the least such n . If f x) = p x) is an irreducible polynomial (other than x) with d[f] = n then ord(/) must divide pTi i There are two important theorems concerning the orders of prime factors and products of relatively prime polynomials over Fg ... 

The set of all polynomials over J g which satisfies the property that any two polynomials, fi x) and f2 x), are equal if and only if fi x) — f2 x) = aa x), where a JFq, constitutes a ring called the ring of polynomials over J-g modulo a x). The ring of polynomials over J g modulo p x), where p x) is an irreducible polynomial, is also a field. If d p] = k, then this field is represented by the set of all p polynomials of degrees fc—1 or less over J g and is called the Galois field of order p. Every finite field J-g is isomorphic (be. can be put into a one-to-one correspondence) with some... 

Bt i.s generally true that over the finite field of size p, where p is some prime, —1 =, ... 

Exact calculations have already been carried out for simple one and two dimensional Euclidean geometries by exploiting properties of polynomials (chapter 5.2.1) and circulant matrices (chapter 5.2.2) over the finite field J-[q, q p wherep is prime. We will here rely instead on the theory of input-free modular systems, which is more suitable for dealing with the dynamics of completely arbitrary lattices. 

The general problem simplifies considerably in the finite field. F[2. Because circuits are always counted at least twice, their number contributes a factor = 0 (mod 2) we see from equation (5.14), therefore, that the only structural information necessary to obtain Pi x) is that of the parity of disjoint edge distributions. Moreover, since there is no way to distribute disjoint edges among an odd number of vertices, equation (5.13) gives... 

Table 5.4 FVaction of symmetric (0, l)-matrices which are nonsingular in finite fields T p (p-prime) - gives the probability that a random (undirected) lattice evolving according to an OT 0p rule yields reversible behavior.

Iidl86] Lidl, R., H.Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press (1986). 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of Z, i.e. there is a variety Xr defined over R such that Xr xr C = X. For every prime ideal p of R let Xp = XR xr R/p. There is a nonempty open subset U C pec(R) such that Xp is smooth for all p U, and the /-adic Betti-numbers of Xp coincide with those of X for all primes / different from the characteristic of A/p (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R/m is a finite field Fq of characteristic p /, we call Xm a good reduction of X modulo q. 

To compute the Betti numbers of some of these varieties we will again use the Weil conjectures. So we have to count their points over finite fields. First we look at the local situation. Let k be a field and R =. .., x,j]]. As above Hilb"(ii)... 

Lemma 2.5.15. There is a finite field extension Fq of F q such that for all finite extensions Fq of Fq ... 

Eigenvalues of Frobenius acting on algebraic varieties over finite fields, Proceedings of Symposia in Pure Mathematics Vol. 29, Algebraic Geometry, Areata 1974, 231-261. 

---