Hamiltonian expansion

The basic elements of the second-quantization formalism are the annihilation and creation operators (Linderberg and (3hrn, 1973). The annihilation operator ap annihilates an electron in orbital f)p (which we assume real), while the creation operator ap (the conjugate of ap) creates an electron in orbital p. These operators satisfy the anticommutation relations 

(9) the summation is over spin. In atomic units the one- and two-electron Hamiltonian integrals are obtained by integration over electronic coordinates according to 

The integrals are calculated in terms of the atomic orbitals (AOs) and are subsequently transformed to the orthonormal basis. In some cases it may be more efficient to evaluate the expressions in the nonorthogonal AO basis. We return to this problem when we consider the calculation of the individual geometry derivatives. For the time being we assume that the Hamiltonian is expressed in the orthonormal molecular orbital (MO) basis. The second-quantized Hamiltonian [Eq. (8)] is a projection of the full Hamiltonian onto the space spanned by the molecular orbitals j p, i.e., the space in which calculations are carried out. 

So far we have considered the Hamiltonian at one geometry, as appropriate for single-point calculations. However, if we wish to calculate the derivatives of the energy with respect to variations in the geometry, we must also consider the geometry dependence of the Hamiltonian. This introduces certain complications, which are treated in the remainder of this section. 

The Hamiltonian integrals depend on the molecular geometry in two ways. The first is trivial and arises because the Coulomb interactions between the electrons and the nuclei depend on the geometry. The second is more complicated and arises because the orbitals are themselves functions of the geometry. The reason for this is that the MOs are expanded in a finite set of AOs fixed on the nuclear centers. A consequence of using a finite set of AOs is that we are presented with a different basis set at each geometry. 

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We encountered in this case the interesting situation of expressing the PJT and a JT vibronic constants by an unique set of parameters offered by the spin Hamiltonian expansion ... 

Inserting this expansion and the Hamiltonian expansion [Eq. (27)] in the amplitude equations, we obtain to first and second orders... 

Figure 1 MCTDH gain factor. The different curves refer to different numbers of degrees of freedom. The parameters used to generate this plot are c /co = 3, C2/C0 = 50, and N = 100. The gain shown is not universal, because it depends on these parameters. However, the chosen values of the parameters are typical for systems not requiring a very large number, S, of Hamiltonian expansion terms. Note that the gain can become very large (>10 ) if there are four or more degrees of freedom and if the contraction efficiency is sufficiently large...

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of fixed radius p. This boundary condition furnishes the quantum numhers n - and each of which is 2D since the reference Hamiltonian hj has two angular degrees of freedom. The superscripts n(, Q in Eq. (95), with n refering to the union of and indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95). 

If the PES are known, the time-dependent Schrbdinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

The symmetric coupling case has been examined by using Sethna s approximations for the kernel by Benderskii et al. [1990, 1991a]. For low-frequency bath oscillators the promoting effect appears in the second order of the expansion of the kernel in coj r, and for a single bath oscillator in the model Hamiltonian (4.40) the instanton action has been found to be... 

Fig. 5. Band gap as a function of nanotube radius calculated using empirical tight-binding Hamiltonian. Solid line gives estimate using Taylor expansion of graphene sheet results in eqn. (7).

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

The MPn method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MPl, MP2, MP3,. .. converges very slowly. 

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