Born-Oppenheimer hamiltonian

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Later we will discuss conventional Born-Oppenheimer calculations used in conjunction with the current work. Thus, for completeness, we will cover here the integrals needed for these calculations. These integrals are quite similar to the ones used in the non-Bom-Oppenheimer calculaions, as will be shown below. We show first the integrals over the Born-Oppenheimer Hamiltonian ... 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

H being the Hamiltonian operator. Throughout this Chapter, we use the usual non-relativistic Born-Oppenheimer Hamiltonian in which no spin operators occur and the nuclear positions are considered fixed. 

Since spin-orbit coupling is normally not included in the Born-Oppenheimer Hamiltonian, singlet and triplet states can be distinguished. In a discussion of photochemical processes, large areas of the nuclear configuration space are of interest, and it is useful to label the energy surfaces in a way that differs from the one conventionally used by spectmscopists. Ai inv... 

Let us consider in detail, for example, kfK Applying the second-order perturbation approach to the adiabatic Born—Oppenheimer Hamiltonian, H, of a given diatomic molecule, we obtain the following expressions for k(2 (Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Murrell, 1960 Salem, 1963b) ... 

In Eq. (3.2), p, fiv, fiz, fia are dipole moment operators and , E2, 3, E4 are the temporal field amplitudes corresponding to each of the four photons. Formally, H is the full electronic-nuclear Hamiltonian. However, in practice, H is the Born-Oppenheimer Hamiltonian of either the ground or excited electronic state, depending on the resonance condition satisfied by the preceding photons. 

Let 0> be the exact -electron ground state of the Born-Oppenheimer Hamiltonian H for a given atomic or molecular system. Likewise, let X > be some exact excited state of interest for the same system with the same nuclear geometry. The corresponding state energies are denoted and respectively. For excitation energy calculations X> is an excited N -electron state, whereas in ionization potential or electron affinity cases X > is an - l)-electron state or an (A -l- l)-electron state, respectively. The commutator of H with the operator Oj = X><0 is easily evaluated,... 

The Born-Oppenheimer Hamiltonian operator H = (T + V) for the fixed-nucleus approximation (Eq. 2.5) is expanded in powers of the displacement (R — Rg) near Rg and second-order perturbation theory is used to calculate E correct to second-order. Then k is given by... 

Taking the nonrelativistic Born-Oppenheimer Hamiltonian in which the nuclei are in fixed positions and in which there are no spin operators, we can write the exact stationary state wave function in the form [97]... 

Then /) satisfies an eigenvalue equation and i R ) is an eigenstate of the so-called Born-Oppenheimer Hamiltonian, provided that that the electronic state r, R ) is so weakly parametrized by the nuclear coordinates that. 

The Born-Oppenheimer Hamiltonian is widely used in solid state physics and quantum chemistry to study the electronic properties of materials - and it is also widely used in this book. In the next section we recast it in the very convenient second quantization representation. 

The Born-Oppenheimer Hamiltonian describes the electronic degrees of freedom. A convenient representation of fermion Hamiltonians is by second quantization. As this representation is widely used in this book, we give a brief discussion of it here. A good discussion may be found in (Landau and Lifshitz 1977) or (Surjan... 

Using these rules, it can be shown that in second quantization the Born-Oppenheimer Hamiltonian is expressed as,... 

Now, the second quantization representation of the Born-Oppenheimer Hamiltonian, eqn (2.19), is valid for an orthonormal basis. Since the atomic orbitals are not automatically orthonormal, they must first be orthogonalized before they... 

With these approximations, we may write a highly simplified Born-Oppenheimer Hamiltonian for the 7r-electrons as. 

If the chains are weakly coupled we may regard H as a perturbation on the approximate Hamiltonian, Hq, where Hq contains the intramolecular Hamiltonians and the remaining interchain interactions - particularly the Coulomb interactions. Thus, we may write the Born-Oppenheimer Hamiltonian for a pair of coupled polymer chains as,... 

Figure 7.18 Eigenvalues and E, (/ =0, 1, 2,...) of the Born-Oppenheimer Hamiltonian, solid lines eigenvalues E n = 0, 1, 2,...) of the perturbed Hamiltonian, dashed lines. The latter eigenvalues are obtained from the graphical solutions in Fig. 7.17. Note the general property that there is one perturbed eigenvalue between every pair of Born-Oppenheimer levels.

The Second Quantized Form of the Born-Oppenheimer Hamiltonian... 

The hydrogen molecule ion, hJ, is the simplest possible molecule. It consists of two nuclei and a single electron, as depicted in Figure 20.2. It is highly reactive, but it is chemically bonded and has been observed spectroscopically in the gas phase. We apply the Born-Oppenheimer approximation and place our coordinate system with the nuclei on the z axis and the origin of coordinates midway between the nuclei. One nucleus is at position A and the other nucleus is at position B. The Born-Oppenheimer Hamiltonian for the hydrogen molecule ion is... 

We can omit the constant from the electronic Hamiltonian and later add that constant to the energy eigenvalues (see Problem 15.6). The Born-Oppenheimer Hamiltonian is now... 

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