Change of Origin

1 Change of Origin. - The vector potential giving rise to a uniform field can be written as 

If the origin of co-ordinates is displaced through a constant vector, Tq, then 

Here Ao is a constant quantity which does not vary with the electron co-ordinates in the new co-ordinate system. It is therefore possible to write the quantity, f as / = Afl.r and the transformed solution of Schrodinger s equation as 

Early calculations of the magnetizability of small molecules, often using methods adapted to the particular case, are due to Pople, Ransil, Karplus and Kolker, Stevens, Pfizer and Lipscomb, Hameka. References can be foimd in Davies and in the earlier Specialist Periodical Report by Hinchliffe and Boirnds. Earlier work using LAOs has been extensively reviewed by Ditchfield.  

While the LAOs are now a well established ingredient of most computational procedures, the original interpretation of the properties of aromatic molecules in terms of ring currents has often been challenged (see for example Bilde and Hansen and other references later in the review). 

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The skew-symmetric part S 4 is equivalent to a vector (x t)/2 with components (/. t),/2 = (/.jtk — /.ktj)/2, involving correlations between a libration and a perpendicular translation. The components of S 4 can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts p, are... 

In the case of an interaction like O + OH, the center of geometry of the diatomic molecule may still be adopted as the reference provided one introduces the modifications connected with the change of origin. For example, if we define (r, 9) and (r 0 ) as the coordinates of the center of mass and center of geometry, respectively, and if z represents the distance between those centers, then the change from one coordinate system into the other will involve the following changes ... 

By suitable change of origin it is possible to eliminate terms in x and y from the general equation of the second degree, (8), 6.VIIIN, and by rotating the axes through a suitable angle it is possible to eliminate the term in xy. Consider the equation ... 

Change of origin of the reference system also induces a gauge transformation and, consequently, leads to alternative partitions of tensors (93) and (99). They will be examined in some detail in Section VI. 

Some molecular tensors (electric dipole polarizability, electric and magnetoelectric shielding) are origin independent, as can be immediately found by inspection of definitions (87)-(112). Other tensors depend on the origin assumed for the multipole expansion. For instance, in a change of origin... 

Rotation of the coordinate system removes the crossproduct terms from the model. A change of origin to the stationary point removes the linear terms. It is obvious that any conclusions as to the nature of the stationary point are reasonable only if the stationary point is within or in the close vicinity of the explored domain. However, it is often found that the stationary point is remote from the design center and that the constant js in the canonical model corresponds to a totally unrealistic response value, e.g. a yield > 100 %. It may also occur that the experimental conditions at the stationary point are impossible to attain, e.g. they may involve negative concentrations of the reactants. Under such circumstances, the response surface around the stationary point does not represent any real phenomenon. It should be borne in mind that a polynomial response surface model is a Taylor expansion of an underlying, but unknown, "theoretical" response function, =... 

The electron coordinates change to 7 == f — G, the nuclear coordinates to Rgc = Rg — G, while the electron coordinates fg relative a given nucleus remain the same. This means that the spin-spin coupling tensor is invariant to the change of origin. 

In these equations a, a, b, b, Sq, and Sq are the fitting parameters, of which only the last two will be used. The value obtained for Sq represents the location along the reaction coordinate of the equilibrium structure of NH3, Then s is converted to s by the following change of origin ... 

The structure semi-invariants are single phases, or linear combinations of phases, which are invariant only for a permitted change of origin. The origin... 

It Should be realised that the multipole moments are not generally invariant with respect to change of origin. The moments Qlj (S) evaluated with respect to an origin at S are related to the moments Qjj,(S+a) with respect to an origin at S+a by[5,8]... 

It is only for free atoms, however, that there is a unique natural origin, the nucleus, around which the angular momentum may be quantized. More generally, H ag in (11.3.12) contains A = BXr, and is thus origin-dependent the vector potential vanishes at the arbitrary origin of coordinates (r = 0, A = 0), but our results must be invariant, as indicated in Section 11.1, under change of origin and indeed under... 

Finally, we note that all the results of this section are expressed in terms of current densities (/,/ ,/ " ), which are defined in a gauge-invariant manner. Under a change of origin, the density matrix P must be changed according to (11.1.20) and the current density (and its component parts 7°, / " ) will be unaffected as a result. 

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