Terms Linear in

The slow convergence of the doubles contributions to the AE is a general problem related to the accurate description of electron pairs in any electronic system. This problem has been studied carefully for the simplest two-electron system, namely the ground-state He atom. In nonrelativistic theory, its Hamiltonian reads 

Highly Accurate Ab Initio Computation of Thermochemical Data 

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

To see the importance of the ri2 term, consider the standard FCI expansion of the He ground-state wavefunction. The FCI wavefunction is written as a linear expansion of determinants, 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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To the same order of approximation of the equations, that is, with only terms linear in / (v) kept, better approximations to the viscosity may be found by considering the equations of higher order than Eqs. (1-86) and (1-87). These new equations will, to this order of approximation, have zero on the left sides (since the higher order coefficients are taken equal to zero) on the right sides appears the factor (p/fii) multiplied by a series of terms like those in Eq. (1-110). Using these equations, and the first order terms of Eq. (1-86) for arbitrary v,... 

Focussing on terms linear in the applied field B, the induced magnetic field at the field point R obtains as the expectation value of B "(R, Ro,B) with respect to the first order wave function corresponding to eq.(6), yielding... 

It should be noted that, in this notation, the order of the superscripts is irrelevant ab... is the entire perturbed energy term linear in ab. .. z and there is no... 

To calculate the liquid-liquid coexistence curves, we cast Eq. 5 in the standard Flory-Huggins form where all terms linear in 0 are subtracted. The free-energy expression then becomes... 

Importantly, from Eq. (A 1.51) it follows (q=0) = 0 (in view of Eq. (A 1.38)) and hence Eq. (2.50) yields symmetry properties expressed by Eq. (A 1.53) result in the fact that there are no terms linear in q in the expansion of <3> (q) in small q. The expansion will, therefore, begin with quadratic terms which have the following general form for an isotropic lattice ... 

In the expansion (A2.32), the first term is merely a constant, while the second one renormalizes equilibrium atomic positions but gives no contribution to the interaction of the atom C with a thermostat (provided a symmetric disposition of atoms, the term linear in r vanishes). The third term contains small corrections to... 

For computational simplicity the series is usually truncated after the terms linear in Sx. Then... 

Using only terms linear in the Casimir operators. 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

Using only terms linear in the Casimir operators. c Using all the terms bilinear in the Casimir operators in Eq. (4.92). 

From his first paper (Mulliken 1925a), Mulliken understood that the band heads did not represent a transition from a non-rotating initial state to non-rotating final state. Yet, he used the band heads to study the vibrational isotope effect since he could measure the band heads more easily and since the rotational energy differences are very small compared to the vibrational energy difference. From the theory, the terms linear in n and n" (ain and bin") arise from the harmonic approximation with the coefficients ai and bi corresponding to the harmonic vibrational frequencies in the... 

The terms linear in A appear now on the diagonal positions of the transformed Hamiltonian D... 

The paramagnetic contribution can again be defined as a response function involving the terms linear in A. We write it as a sum of six terms... 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

Kutzelnigg, W., Klopper, W. Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory. J. Chem. Phys. 1991, 94, 1985-2001. 

You may neglect the term proportional to 2 merely obtain the term linear in e. 

Terms linear in the large logarithm were recently obtained in [80, 81]... 

We now see that for a real proton the charge radius contribution has exactly the form in (6.3), where the charge radius is defined in (6.7). The only other term linear in the momentum transfer in the photon-nucleus vertex in (6.9)... 

If this result is substituted into Equation (11), expanded, and only terms linear in dr retained, the expression becomes... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

As an example of the connection between perturbation theory wave function corrections and polarizability, we now calculate the linear polarizability, ax. The states are corrected to first order in H. Since the polarization operator (Zx) is field independent, polarization terms linear in the electric field arise from products of the unperturbed states and their first-order corrections from the dipole operator. The corrected states are [12]... 

Together with the ansatz Eq. (1.1), Eq. (1.5) describes the response of a liquid film to an applied pressure p. The resulting differential equation is usually solved in the limit of small amplitudes q < h hv and only terms linear in f are kept ( linear stability analysis ). This greatly simplifies the differential equation. The pressure inside the film p = Pl + Tex consists of the Laplace pressure pL = —ydxxh, minimizing the surface area of the film, and an applied destabilizing pressure pex, which does not have to be specified at this point. This leads to the dispersion relation... 

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