Magnetic, Electric, and Mechanical Properties of Polymers

Abstract

If we neglect interaction of the electronic and/or nuclear spins with an external magnetic field (which can be treated correctly only within the framework of a relativistic CO theory; see Section 1.5), the one-electron Hamiltonian of a system in the presence of a magnetic field H can be written in the form

$$\overset{x}{\mathop{H}}\,=\frac{\overset{\wedge }{\mathop{p}}\,}{2m}+V(r)$$

(10.1)

where

$$\overset{\wedge }{\mathop{p}}\,=\overset{\wedge }{\mathop{p}}\,-\frac{e}{c}A;\,H(r)=curlA(r)$$

(10.2)

In the Hartree-Fock case one can then write

$$\overset{\wedge }{\mathop{F}}\,=\overset{x}{\mathop{{{H}^{N}}}}\,+\sum\limits_{j}{(2\overset{\wedge }{\mathop{{{J}_{j}}}}\,-\overset{\wedge }{\mathop{{{K}_{j}}}}\,)}$$

(10.3)

with

$$\overset{x}{\mathop{{{H}^{N}}}}\,=\frac{1}{2m}{{(\overset{\wedge }{\mathop{p}}\,-\frac{e}{c}A)}^{2}}-\sum\limits_{{{q}_{1}}=-N}^{N}{\sum\limits_{\alpha =1}^{M}{\frac{{{Z}_{\alpha }}}{|r-R_{\alpha }^{{{q}_{1}}}|}}=\frac{\overset{\wedge }{\mathop{p}}\,}{2m}+EN}$$

(10.4)

for a linear chain of 2N + 1 unit cells and M atoms in the cell, where the abbreviation EN denotes all the electron—nuclear interaction terms in the chain. In equation (10.3) summation over j indicates, as before, summation over all filled bands and integration in k-space over the first Brillouin zone.