Hamiltonian in the new coordinates

The new coordinates have to be introduced into the Hamiltonian. To this end, we need the second derivative operators in the old coordinates to be expressed by the new ones. First (similarly as in Appendix I), let us construct first derivative operators  

For the first derivative operator with respect to the coordinates of the electron i we obtain  

After inserting all this into the Hamiltonian (6.1) we obtain the Hamiltonian pressed in the new coordinates  

The Hq does not contain the kinetic energy operator of the nuclei, but all the other terms (this is why it is called the electronic Hamiltonian) the first term stands for the kinetic energy operator of the electrons, and V means the potential energy corresponding to the Coulombic interaction of all particles. The first term in the 

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Other coordinates do not change. The full Hamiltonian in the new coordinates reads... 

The transformed interaction Hamiltonian Hsb of Eq. 15.6 and the bath Hamiltonian of Eq. 15.8 define the system-bath Hamiltonian in the new coordinates. The subsystem part, comprising the electronic subspace and possibly a subset of strongly coupled vibrational modes, has remained unchanged. 

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

Dropping the centre of mass part, the Hamiltonian (5) in the new coordinates becomes... 

The procedure of transforming the Laplacians from the ones expressed in the old coordinates to those given in the new coordinates (the same one shown in Appendix 1 available at booksite.elsevier.com/978-0-444-59436-5), leads to three mutually independent Hamiltonians ... 

In the new coordinates, the bath Hamiltonian takes a hierarchical form The effective modes Xg couple directly to the electronic subsystem, while the remaining (residual) Nb —Ngs bath modes couple in turn to the effective modes. The new bath Hamiltonian Hb of Eq. 15.8 can thus be split as follows ... 

Transform the Hamiltonian to the normal form described above up to the desired degree of accuracy using a symbolic manipulator. The Hamiltonian is now in a new coordinate system that we will call the normal form coordinates. ... 

The transformation from one pair of canonically conjugate coordinates q and momenta p to another set of coordinates Q = Q(p,q,t) and momenta P = P(p>qT) is called a canonical transformation or point transformation. In this transformation it is required that the new coordinates (P,Q) again satisfy the Hamiltonian equations with a new Hamiltonian H P,Q,t) [35] [43] [52]. 

The configuration coordinates of electrons (p) and nuclei (R) in the new frame are related to the laboratory one by rk = u + pk, Qk. = u + Rk, symbolically written as r =u+p, Q=u+R, and T=(p,R). Ke represents the electrons kinetic energy operators Vee (p), VeN(p, R) and Vnn(R) are the standard Coulomb interaction potentials they are invariant to origin translation. The vector u is just a vector in real space R3. Kn is the kinetic energy operator of the nuclei, and in this work the electronic Hamiltonian He(r Z) includes all Coulomb interactions. This Hamiltonian would represent a general electronic system submitted to arbitrary sources of external Coulomb potential. 

The Hamiltonian is also invariant with respect to any rotation in space U of the isotropy of coordinate stem about a fixed axis. The rotation is carried out by apptying an or-thogonal matrix transformation V of vector r = (x, y, z) that describes any particle of coordinates x, y, z. Therefore all the particles undergo the same rotation and the new coordinates are r = Ur = Ur. Again there is no problem with the potential energy, because a rotation does not change the interparticle distances. 

After the transformation, the Hamiltonian is expressed in terms of the new coordinates q,p). ... 

The total effective Hamiltonian H, in the presence of a vector potential for an A + B2 system is defined in Section II.B and the coupled first-order Hamilton equations of motion for all the coordinates are derived from the new effective Hamiltonian by the usual prescription [74], that is. 

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

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