Finite fields, other

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 ... 

An alternative approach is to apply stronger fields and only use energies calculated for positive field strengths in generating the polynomial fit. In this case the energy is a function of both odd and even powers in the polynomial fit. We will show that the dipole moments derived from our non-BO calculations with the procedure that uses only positive fields and polynomial fits with both even and odd powers match very well the experimental results. Thus in the present work we will show results obtained using interpolations with even- and odd-power polynomials. Methods other than the finite field method exist where the noise level in the numerical derivatives is smaller (such as the Romberg method), but such methods still do not allow calculation of odd-ordered properties in the non-BO model. 

Much information of interest for atomic and molecular systems involves properties other than energy, usually observed via the energy shifts generated by coupling to some external field. The desired property is then the derivative of the energy with respect to the external field, which may be obtained by two different approaches. The finite-field method solves the Schrodinger equation in the presence of the external field, yielding... 

Dickson and Becke59 performed finite-field LDA calculations of the dipole polarizabilities and hyperpolarizabilities of the following compounds H2, N2, 02, CO, HF, H20, NH3, and CH4. These studies have a benchmark character (for dipole polarizabilites and first hyperpolarizabilities). The calculated dipole polarizabilities are systematically overestimated (see Table 2-3). Other studies reveal the similar trend that LDA overestimates the dipole polarizabilities of small organic molecules. 

A system in which an addition and a multiplication are defined and for which the commutative, associative, and distributive properties hold, where there exist identities for addition and multiplication, where every element has an additive inverse, and every non-zero element has a multiplicative inverse, is called a field. Other examples of fields are the set of rational numbers and the set of real numbers. Since the number of elements in our set is finite, we have an example of finite field. 

The perturbed total energies or other properties of the system can be written as an expansion in terms of moment and polarizability components (see Section I). If different values of the field strength or charge positions are used, a system of simultaneous equations can be written from the truncated series, and these equations are solved to find the unknown polarizabilities. The system of equations must be chosen sufficiently large to ensure that the truncation error is minimized, but sometimes it is not practical to carry out the number of finite-field calculations that this might call for. 

Other CCSD calculations show considerable variations especially with the level of correlation (see Kongsted et al.54, Jensen49) but are generally within about 20% or better of the experimental value. The Maroulis42 finite field static values with very large basis sets and CCSD(T) correlation are still an important... 

Maroulis et al.122 have applied their static polarizability, finite field technique to a study of the 22 electron diatomics CP , BC1, CC1+ and PO+. The vibrational contribution to the ground state polarizability has also been calculated. The dipole polarizability and other properties of YbF have been investigate in the unrestricted Dirac-Fock approximation by Parpia123 and the static second hyperpolarizability of the Cu2 dimer has been calculated in a correlation corrected UHF study by Shigemoto et al.124... 

The authors point out a few important aspects accompanying their calculations of the halogen field gradients also being relevant for other systems. At first the applied perturbation strength in the finite field treatment for the EFG claculation according to... 

This section does not provide an exhaustive survey of all possible methods and approximations in use, but rather focuses on the most common methods currently implemented and then briefly describes a few others that are important. Three methods—the finite field, sum-over-states (SOS), and time-dependent Hartree-Foclc methods—encompass the vast majority of NLO property calculations being performed today and are implemented in several readily available molecular orbital computer programs. 

Some types of frequency-dependent quantity can be calculated from finite field procedures, provided they are based on frequency-dependent properties obtained by other methods, such as those discussed later in the chapter. For example, a computer program that calculates the frequency-dependent polarizability can be modified to do so in the presence of a static electric field to yield pEOPE. yOKE values via the expansion- ... 

To apply the method based on the expectation value we need a wave function which satisfies the Hellman-Feynman theorem. This is fulfilled for variational wave functions in which all parameters are fully optimized. Such calculations are possible for small molecules only. For the molecular case, generally applicable and straightforward is the so called finite field method. In the usual experimental situation a molecule is somehow placed in the external field, and the external field will influence the internal field. We can take this into account by expressing the energy of the system as a function of the filed. The finite filed method exploits the expansion of the energy with respect to an electric filed E (or for some properties, also other external fields) along the i -direction. 

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