Z-vector method

These equations can be derived by differentiating the expression for Emp2 and using the Z-vector method to avoid solving the CPHF equations explicitly for each of the derivatives of the MO coefficients. 

The block of vacant-occupied CPHF coefficients f/ of Eq. (2.20), which depend on the specific perturbation a, may be eliminated by using the interchange (Z-vector) method of Handy and Schaefer [20], properly extended to the PCM framework [21] ... 

MO basis y/s and are matrix elements of the one and two-particle density matrices, respectively [16] y / and y/f are matrix elements of the transition one-particle transition density matrix see footnote 5. The differentiated MO integrals involve derivative of the MO coefficients, which can be avoided by solving, or by exploiting the PCM-Z-vector method [17]. 

PCM coupled-peiturbed Hartree-Fock equations (PCM-CPHF), 17 PCM Hartree-Fock equations, 6 PCM Hartree-Fock matrix, 6, 7 PCM Z vector method, 18 Perturbation theory for the energy (PTE), 8, 10, 11... 

Pulay s paper opened the way for analytic second and higher derivatives of the SCF energy. Earlier papers had suggested that this might be prohibitively expensive [7], but the development of an efficient method to solve the couple perturbed HF (CPHF) equations, made the calculation of SCF second derivatives practical [8]. As a consequence, vibrational force constants and frequencies could be calculated routinely and efficiently. Third and fourth geometric derivatives of the SCF energy followed after a few years [9-12]. The solution of the CPHF equations (in their full or reduced Z-vector form [13]) also made post-SCF first derivatives practical and cost-effective. 

The method of constructing the wavefunctions as a linear combination of the slater determinants is discussed in standard textbooks [18]. The spatial wave-functions are constructed in such a way that they transform exactly identical to the x,y, and z vectors in the O group. Following Sugano, Tanabe, and Kamimura [18], the wavefunctions so constructed are labeled as a, p, and y. Usually, the wavefunctions are further constructed by taking a linear combination of these in such a manner that they are the eigenfimctions of the spin operator S. In the case of the triplet state, the Mj values (i. e. eigenvalues for the spin operator SJh) should be +1,0, or -1. Thus, we should have 3x3 = 9 wavefunctions. The nine wavefunctions thus constructed are as follows ... 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. 

In light of the special structure of the diagonal matrices k, whose blocks are lower triangle, the components s = 1,2,..., z, of the unknown vector yjQ.) are to be determined successively by the elimination method in passing from a to qH- 1 and from s to s- 1. By the elimination formulas for a three-point equation we constitute in a term-by-term fashion the vectors a = 1,2,..., p. Moving in reverse order from a -f 1 to a and from s -b 1 to s the vectors y(p+i), >y(2p) recovered from the system... 

Preprocessing is the operation which precedes the extraction of latent vectors from the data. It is an operation which is carried out on all the elements of an original data table X and which produces a transformed data table Z. We will discuss six common methods of preprocessing, including the trivial case in which the original data are left unchanged. The effects of each of these six types of preprocessing will be illustrated numerically by means of the small 4x3 data table from the study of trace elements in atmospheric samples which has been used in previous sections (Table 31.1). The various effects of the transformations can be observed from the two summary statistics (mean and norm). These statistics include the vector of column-means m and the vector of column-norms of the transformed data table Z ... 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

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