Electronic function

The necessary material to perform annual verification of flaw detector is mainly composed of an electronic function generator, an external calibrated attenuator and an oscilloscope. 

Both terms on the right are related to the rate of change of the adiabatic electronic functions with respect to the nuclear coordinates. The first term Gy is given by... 

The diabatic electronic functions are related to the adiabatic functions by unitary transformations at each point in coordinate space... 

Note that, although there is a r esemblance, this ansatz is quite differ ent from the Born representation of Eq. (A.3) due to the time dependence of the electronic functions. By taking a single configuration. 

The basis consisting of the adiabatic electronic functions (we shall call it bent basis ) has a seiious drawback It leads to appearance of the off-diagonal elements that tend to infinity when the molecule reaches linear geometry (i.e., p 0). Thus it is convenient to introduce new electronic basis functions by the transformation... 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

This algebra implies that in case of Eq. (111) the only two functions (out of n) that flip sign are and because all in-between functions get their sign flipped twice. In the same way, Eq. (112) implies that all four electronic functions mentioned in the expression, namely, the jth and the (j + 1 )th, the th and the (/c -h 1 )th, all flip sign. In what follows, we give a more detailed explanation based on the mathematical analysis of the Section Vin. 

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

Any set of one-electron functions can be a basis set in the LCAO approximation. However, a well-defined basis set will predict electronic properties using fewer terms than a poorly-defined basis set. So, choosing a proper basis set in ab initio calculations is critical to the reliability and accuracy of the calculated results. 

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The next approximation involves expressing the jiiolecular orhiiah as linear combinations of a pre-defined set of one-electron functions kjiown as basis functions. These basis functions are usually centered on the atomic nuclei and so bear some resemblance to atomic orbitals. However, the actual mathematical treatment is more general than this, and any set of appropriately defined functions may be u.sed. 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

Without introducing any approximations, the total (exact) wave function can be written as an expansion in the complete set of electronic functions, with the expansion coefficients being functions of the nuclear coordinates. 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Since the exact density matrix is not known, the (approximate) density is written in terms of a set of auxiliary one-electron functions, orbitals, as... 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), <, which are orthogonal (Section 4.2). 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

Let us now consider a system of N electrons, where N+ electrons occupy spin orbitals of a character or plus spin, and N electrons occupy spin orbitals of character or minus spin. By using the separation of the one-electron functions y>k x) into two groups having different spins, we may write the fundamental invariant (Eq. 11.41) in the form... 

The method of superposition of configurations described in the first paper (1928) implies that, after choosing a complete basic set of one-electron functions rpk rx) one can develop the space function in Eq. III. 1 in the form ... 

The method of superposition of configurations as well as the method of different orbitals for different spins belong within the framework of the one-electron scheme, but, as soon as one introduces the interelectronic distance rijt a two-electron element has been accepted in the theory. In treating the covalent chemical bond and other properties related to electron pairs, it may actually seem more natural to consider two-electron functions as the fundamental building stones of the total wave function, and such a two-electron scheme has also been successfully developed (Hurley, Lennard-Jones, and Pople 1953, Schmid 1953). 

Every normalizable antisymmetric wave function can be expressed as the sum of a series of Slater determinants built up from a complete basic set of one-electron functions. 

The results in Table VI are of considerable interest also for atoms with more than two electrons, since they show the possibilities and limitations of the method of superposition of configurations/ when the latter are built up from one-electron functions... 

We note that it is possible to combine the method with correlation factor with the method using superposition of configurations to obtain any accuracy desired by means of comparatively simple wave functions. For a very general class of functions g(r12), one can develop the quotient (r r2)lg(r12) according to Eq. III.2 into products of one-electron functions y>k(r), which leads to the expansion... 

Hurley, A. C., Lennard-Jones, J., and Pople, J. A., Proc. Roy. Soc. [London) A220, 446, The molecular orbital theory of chemical valency XVI. A theory of paired electrons in polyatomic molecules." Use of two-electron functions T fa, x5) as a basis. 

One-electron function, 228, 261 One particle scheme, 210 Ordered configuration, 261 Ortho-iodo-phenol, radiation resistance, 196... 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

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