Tail cancellation

We now place the orbitals at Q and Q , the positions of the two atoms, and write the total wave function as the linear combination 

With the addition theorem (8.7) the cancellation required will take place if 

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As discussed by Andersen [9, 10] for muffin-tin orbitals, the locally regular components y defined in each muffin-tin sphere are cancelled exactly if expansion coefficients satisfy the MST equations (the tail-cancellation condition) [9, 384], The standard MST equations for space-filling cells can be derived by shrinking the interstitial volume to a honeycomb lattice surface that forms a common boundary for all cells. The wave function and its normal gradient evaluated on this honeycomb interface define a global matching function %(cr). 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X> the interior component of 4> Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts... 

The seeds for the development of linear methods may be found in the 1971 paper by Andersen [1.21] which contains a definition of muffin-tin orbitals, an addition theorem for tails of partial waves, and the tail cancellation theorem. Soon after, these ideas were developed into a practicable band-calculation method, the linear combination of muffin-tin orbitals (LCMTO) method [1.22,... 

By means of the one-centre expansion (2.4) of the Bloch sum of MTO tails, the required tail cancellation is seen to occur if... 

If we approximate the crystal potential by an array of non-overlapping muffin-tin wells as in (5.2), the energy-dependent muffin-tin orbitals (5.13) may be used in conjunction with the tail-cancellation theorem to obtain the so-called KKR equations. These have the form (1.21) and provide exact solutions for muffin-tin geometry. Computationally, however, they are rather inefficient and it is therefore desirable to develop a method based upon the variational principle and a fixed basis set, which leads to the computatinal-ly efficient eigenvalue problem (1.19). 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

The LMTO method has the computational speed and flexibility needed to perform calculations of electron states in molecules and compounds. Therefore in the present chapter we shall generalise the LMTO formalism purely within the atomic-sphere approximation to include the case of many inequivalent atoms per cell. The LMTO method is based on the variational principle in conjunction with energy-independent muffin-tin orbitals but, in addition to this approach, we have also considered the tail-cancellation principle which led to the KKR-ASA condition (2.8). Since the latter has conceptual advantages, we apply the tail-cancellation principle to the simplest possible case of more than one atom, namely the diatomic molecule. After that, we turn to crystalline solids and generalise or sometimes rederive the important equations of LMTO formalism. Hence, in addition to giving the LMTO equations for many atoms per cell, the present chapter may also serve as a short and compact presentation of the crystal-structure-dependent part of LMTO formalism. The potential-dependent part is treated in Chap.3. In the final sections are listed the modifications needed to calculate ground-state properties for materials with several atoms per cell. 

To introduce the subject of many atoms per cell, we apply the tail-cancellation theorem, Sect.2.1, to a collection of atoms. In the derivation it is convenient to consider the simplest case, i.e. a diatomic molecule, but the results will be valid for any molecule or cluster. Our starting point is the energy-independent muffin-tin orbitals (2.1) in the atomic-sphere approximation, i.e. 

To correct for this overcounting, we cancel out the number of ways n j heads can be permuted and the number of ways Uj tails can be permuted. Using the same logic as in item (3), these redundant possibilities are given by n j and nj , respectively. Dividing the result in item (4) by these factorials gives Eq. (1.21). 

Three LMTO envelopes were used with the tail energies -0.01 Ry, -1 Ry and -2.3 Ry. In the first two of them, s,p,d orbitals were included and in the last one only. s and p were used. It was necessary to treat the Ti 3p and 3-s states in the semi-core state, i.e. to do a so called 2-panel calculation. The basis set for the second panel consisted of 3-s, 3p, 3d orbitals on the Ti sites and 3-s, 3p orbitals on the Si sites. The same quality k-mesh was used in all calculations to ensure maximum cancellation of numerical errors and to obtain accurate energy differences. 

The structure of the dimers from mero-substituted derivatives was initially determined by comparison of the observed and calculated dipole moments. For a head-to-tail dimer the dipole moments resulting from the 9 and 9 substituents should cancel each other and the resultant dipole moment should be essentially zero. For a head-to-head arrangement the dipoles would be in the same direction and the resultant should be considerably greater than zero. The dimers produced upon irradiation of 9-chloro and 9-bromoanthracene solutions were observed to be 0.36 and 0.60 D, respectively. Since these values are much less than expected for a head-to-head arrangement for these derivatives (3.8 D), it was concluded that both of these dimers were formed in a head-to-tail configuration/30 ... 

To improve our model still further, we have to visualize s- and p-orbitals as waves of electron density centered on the nucleus of an atom. Like waves in water, the four orbitals interfere with one another and produce new patterns where they intersect. These new patterns are called hybrid orbitals. The four hybrid orbitals are identical to one another except that they point toward different comers of a tetrahedron (Fig. 3.16). Each orbital has a node close to the nucleus and a small tail on the other side where the s- and p-orbitals do not completely cancel. These four hybrid orbitals are called sp3 hybrids because they are formed from one s-orbital and three p-orbitals. In an orbital-energy diagram, we represent the hybridization as the formation of four orbitals of equal energy intermediate between the energies of the s- and /7-orbitals from which they are constructed (43). The hybrids are colored green to remind us that they are a blend of (blue) s-orbitals and (yellow) p-orbitals. 

The basic idea of the mufiin-tin theory was then to construct tight-binding combinations -as in Eq. (20-1) -of muffin-tin orbitals, and to require that the tails of all neighboring muffin-tin orbitals cancel the —/l(/-/r, ,) that had been added to the central sphere the result is a solution of the Schroedinger equation everywhere, but the calculation has been reduced to a calculation within a single atomic sphere. The form of the condition that the —/ ( // ,) be cancelled is the same as the LCAO equation, Eq. (20-3), but with the replaced by... 

For most applications, the phase anomalies that are created by zero-quantum coherence can be ignored in practice, because the long dispersive tails of the antiphase components have opposite signs and tend to cancel each other (Ranee, 1987). However, the zero-quantum terms must be suppressed if TOCSY spectra with pure two-dimensional in-phase absorp-... 

A number of so-called double ion-pair methods have been described for the analysis of hydrophobic amines in which the mobile phase contains both a quaternary ammonium ion and an alkyl sulfate or sulfonate. At first glance, this combination of mobile phase additives is counterintuitive because one would expect the effect of the anionic and cationic additives to cancel. However, the combination of cationic masking agents to reduce peak tailing and an anionic ion-pairing agent to enhance retention is sometimes necessary for the reversed-phase separation of hydro-phobic amines. 

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