The Molecular Electronic Hamiltonian

Combining the results of Sections 1.4.1 and 1.4.2, we may now construct the full second-quantization representation of the electronic Hamiltonian operator in the Bom-Oppenheimer approximation. Although not strictly needed for the development of the second-quantization theory in this chapter, we present the detailed form of this operator as an example of the construction of operators in second quantization. In the absence of external fields, the second-quantization nonrelativistic and spin-free molecular electronic Hamiltonian is given by 

Here the Z/ are the nuclear charges, r/ the electron-nuclear separations, ri2 the electron-electron separation and Ru the intemuclear separations. The summations are over all nuclei. The scalar term (1.4.42) represents the nuclear-repulsion eneigy - it is simply added to the Hamiltonian and makes the same contribution to matrix elements as in first quantization since the inner product of two ON vectors is identical to the overlap of the determinants. The molecular one- and two-electron integrals (1.4.40) and (1.4.41) may be calculated using the techniques described in Chapter 9. 

The form of the second-quantization Hamiltonian (1.4.39) may be interpreted in the following way. Applied to an electronic state, the Hamiltonian produces a linear combination of the original state with states generated by single and double electron excitations from this state. With each such excitation, there is an associated amplitude hpg or gpgps, which represents the probability of this event happening. These probability amplitudes are calculated from the spin orbitals and the one- and two-electron operators according to (1.4.40) and (1.4.41). 

In the last three sections we have considered the effect of a time-dependent external electric field r,t) and a magnetic induction B r,t) on the motion of an electron and denoted the corresponding potentials with 4 r,t) and A r,t). In the present section we want to collect all the terms and derive our final expression for the molecular electronic Hamiltonian. However, we will not restrict ourselves to the case of external fields because in the following chapters we want to study also interactions with other sources of electromagnetic fields such as magnetic dipole moments and electric quadrupole moments of the nuclei, the rotation of the molecule as well as interactions with field gradients. Therefore, we do not include the superscripts B and on the vector and scalar potential in this section. On the other hand, we will assume that the perturbations are time independent. The time-dependent case is considered in Section 3.9. 

In the previous sections it was shown that in the minimal coupling approximation the vector potential enters the mechanical momentum of electron i 

As we are working within the Born-Oppenheimer approximation the nuclei are fixed in space and there is thus no coupling between the momenta of the nuclei and the vector potential. 

Secondly, terms consisting of the scalar potential times the charges of the particles have to be added to the Hamiltonian. Although we are only interested in the electronic Hamiltonian, one should also add the constant contribution from the interaction of the scalar potential with the nuclear charges. In total, the following terms due to the scalar potential have to be added 

Thirdly, the interaction of the spin of the electrons with magnetic fields is introduced via the Zeeman term of the Pauli Hamiltonian, Eq. (2.96), 

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The main purpose of DFT is to simplify evaluation of the final term in the molecular electronic hamiltonian ... 

We will mostly use second quantization and thereby represent the molecular electronic Hamiltonian in Eq. (13-2) as... 

The over-riding factor which governs the choice of the mathematical form of anything which appears in the molecular electronic Hamiltonian is... 

NMR parameters as energy derivatives Now we define NMR parameters as energy derivatives using the expression for the molecular electronic Hamiltonian in the presence of the magnetic field. 

For the computation of the second derivatives given in Eq. (6.8) for the shielding tensor, it is necessary to specify the molecular (electronic) Hamiltonian H in the presence of a magnetic field. The latter is obtained by replacing the canonical momentum p in the kinetic energy part of H by the kinetic momentum n... 

Now consider the ground-state energy expression. We write the molecular electronic Hamiltonian as the sum of two H-atom Hamiltonians plus perturbing terms ... 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system... 

We now consider the orbital degeneracy of molecular electronic terms. This is degeneracy connected with the electrons spatial (orbital) motion, as distinguished from spin degeneracy. Thus 11 and 11 terms of linear molecules are orbitally degenerate, while and 2 terms are orbitally nondegenerate. Consider an operator P that commutes with the molecular electronic Hamiltonian and that does not involve spin we have... 

Here H is the molecular electronic Hamiltonian, and is the transition density defined as with being the total number... 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system contains parameters. For example, in the one-dimensional harmonic-oscillator Hamiltonian operator - f /2m) (f/dx ) + kx, the force constant is a parameter, as is the mass m. Although is a constant, we can consider it as a parameter also. The stationary-state energies E are functions of the same parameters as H. For example, for the harmonic oscillator... 

Collecting all terms we can finally write the molecular electronic Hamiltonian operator H in the presence of an electromagnetic field as... 

In the previous chapter we have derived the molecular electronic Hamiltonian H in the presence of static electromagnetic fields or fields due to nuclear moments. Throughout this chapter we will consider a general field and denote it as T with components J-a - -. Examples for. Fq,... are one of the three components of the electric field a, of the magnetic induction Ba, of the nuclear moment of a magnetic nucleus K or one of the nine components of the field gradient Sa/3- ... 

The perturbation operators are again the first derivatives of the molecular electronic Hamiltonian... 

In this appendix, explicit expressions for all the perturbation operators are collected. They were derived in Chapters 4 to 8 by expressing the scalar and vector potentials in the molecular electronic Hamiltonian, Eq. (2.101), in terms of electric fields and various magnetic inductions. 

The perturbation operators O are obtained as derivatives of the molecular electronic Hamiltonian, Eqs. (2.101) and (2.108), evaluated for zero fields or magnetic moments ... 

Owing to the presence of the Coulomb potential, the molecular electronic Hamiltonian becomes singular when two electrons coincide in space. To balance this singularity, the exact wave function exhibits a characteristic nondifferentiable behaviour for coinciding electrons, giving rise to the electronic Coulomb cusp condition [3]... 

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