Total energy per unit cell

The distances found between platinum centers in these molecules have been correlated with the resonating valence bond theory of metals introduced by Pauling. The experimentally characterized partially oxidized one-dimensional platinum complexes fit a correlation of bond number vs. metal-metal distances, and evidence is presented that Pt—Pt bond formation in the one-dimensional chains is resonance stabilized to produce equivalent Pt—Pt distances.297 The band structure of the Pt(CN)2- chain has also been studied by the extended Huckel method. From the band structure and the density of states it is possible to derive an expression for the total energy per unit cell as a function of partial oxidation of the polymer. The equilibrium Pt-Pt separation estimated from this calculation decreases to less than 3 A for a loss of 0.3 electrons per platinum.298... 

Among the five parameters three remain free and have to be determined variationally by minimalization of the total energy per unit cell. 

The results of a study of the Huckel Hamiltonian enabled the following classification of the shape of the total energy per unit cell, E, as function of the asyimetric distortion. A, of the chain (i) E 1 1 It yA (Peierls case) (ii)... 

In Eqs. (4), (6) and (11), there are infinite summations in the overlap matrix Sk, the Fock matrix Fk, and the total energy per unit cell E JF. There are also infinite summations in T, V, Jlpq, and Kln. The infinite lattice summations in Sk, Tl and K converge themselves while V , J and the internuclear interactions have to be summed together to get converged results [20-22], In real calculations, cutoffs for the lattice summations have been imposed and multipole expansion techniques have been applied to hasten the convergence [22],... 

Since the total energy is infinite in an extended system, we are more interested in the total energy per unit cell, which is defined as... 

From Raleigh-Schrodinger perturbation theory [3], one can get analytical expressions for any given-order MBPT correction to the total energy per unit cell. Here we focus on the second-order MBPT correction which is given by [26]... 

For polymers, we can get analytical expressions for the convergence of E and 4k with the lattice summation cutoff N. Let us use E (N) and 4k (N) to denote the approximate MBPT(2) corrections to the total energy per unit cell and energy of the nth band with the reciprocal vector k, respectively. With multipole expansion techniques, we have shown that when N is large enough, E (N) is related to the exact value by [27]... 

The energy Euc in Eq. (33) can be E JiPI(2> or other correlated total energy per unit cell. The coordinates Y A are Fourier transforms of the nuclei s Carte-... 

Firstly, we calculated the vector of averaged effective H-bond energies Ue//k) and the matrix of averaged interactions of opposite molecules in hexagonal cycles Uopp(m,n). Total energy per unit cell to third neighbor interaction accuracy is... 

The Calculation of Effective Total Energy per Unit Cell. - The NFC method does not provide, however, an effective total energy per unit cell of a disordered chain. For this purpose one can apply either a cluster method22 or the so-called elongation method.23... 

In the case of a periodic polymer (assuming that all nuclei move in phase with each other in the direction of x), Hirata and Iwata48 have given analytical expressions for the first and second derivatives according to the nuclear coordinates in the HF level, Writing for the total energy per unit cell... 

It should be noticed that the authors did not provide an analytical expression for the second derivative of the total energy per unit cell in the MP2 case. 

In the case of the basis applied one can estimate the valence-shell correlation, which is in its calculation about 70-75%. [The total energy per unit cell at the MP2 level is — 77.168 a.u., at the HF level —76.893 and their difference is 7.5 eV.55 On the other hand the valence shell-correlation energy of an acetylene unit was estimated to be 10 eV56]. Extrapolation to 100% correlation has given a gap of 2.5 eV.M... 

The DFT has been successfully applied in different levels of approximation to the ground state properties of 3D solids and of molecules. Though there are a few DFT calculations for polymers in the literature,80,81 these calculations have not included a systematic investigation of the dependence of their total energy per unit cell on their conformation and of their valence electron structure (photoelectron spectra). 

The total energy per unit cell of the polymer is given (in atomic units) as... 

The k appears as a parameter in the equation similarly to the nuclear positions in molecular Hartree-Fock theory. The solutions are continuous as a function of k, and provide a range of energies called a band, with the total energy per unit cell being calculated by integrating over k space. Fortunately, the variation with k is rather slow for non-metaUic systems, and the integration can be done numerically by including relatively few points. Note that the presence of the phase factors in eq. (3.76) means that the matrices in eq. (3.79) are complex quantities. 

How do we calculate the total energy per unit cell This is a different question. Let us denote this quantity by Ej- Since a crystal only represents a very large molecule, we may use the expression for the total energy of a molecule. In the 3-D case, we get... 

Table 9.3. Total energy per unit cell Ej- in the pofymer LiH as a function of unit cell definition (Fig. 9.23). For each choice of unit cells, this energy is computed in four ways (1) without long-range forces (long range = 0) i.e.. unit cell 0 interacts with N = 6 unit cells on its right-hand side and N unit cells on its left-hand-side (2), (3), (4) with the long range computed with multipole interactions up to the and a terms, respective. The...

Therefore, the total energy per unit cell amounts to... 

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