Derivative Hamiltonians

If the zero-order wavefunction does, in fact, satisfy Eqn. (26), then the first term on the right in Eqn. (32) is identically zero. This means that the first derivatives are obtained as an expectation value of the derivative Hamiltonian. This last statement is the Hellmann-Feynman theorem. 

Basis sets can be employed to solve derivative Schrodinger equations as naturally as employing them to solve the basic Schrodinger equations. An organized way of using basis sets, and a way that is quite suited for computational implementation, is to cast operators into their matrix representations in the given basis. This needs to be done for the zero-order Hamiltonian and for each derivative Hamiltonian operator. The zero-order Schrodinger equation for one state in matrix form is... 

In order to keep track of all terms simply, it is much more convenient to consider our derivative Hamiltonian to be written in second quantization, +... 

This says that the dipole moment operator will be needed for the derivative Schrodinger equations involving derivatives with respect to V (the -com-ponent of a uniform field), or that the second moment operator will be needed for derivative Schrodinger equations involving differentiation with respect to field gradient components such as V. In general, there will be operators combined with parameters in the Hamiltonians, and then the derivative Hamiltonians will be operators of some sort. These must be constructed. 

Group-theoretically derived Hamiltonian See [99Lng]. All valnes inMHz. 

We now show what happens if we set up tire Hamiltonian matrix using basis functions i ), tiiat are eigenfiinctions of Fand with eigenvalues given by ( equation A1.4.5) and (equation Al.4.6). We denote this particular choice of basis fiinctions as ij/" y. From (equation Al.4.3). (equation A1.4.5) and the fact that F is a Hemiitian operator, we derive... 

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

A very simple procedure for time evolving the wavepacket is the second order differencing method. Here we illustrate how this method is used in conjunction with a fast Fourier transfonn method for evaluating the spatial coordinate derivatives in the Hamiltonian. 

Hamiltonian H. Fonnally, at least, these expressions avoid the detennination of scattering wavefrmctions and any infonnation related to the internal states of the reagents or products. The frmdamental derivations in this area have been presented by Miller [44] and by Schwartz et al [45]. 

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

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. 

A formal derivation of the location of the zeros of Cg t) for a general adiabatic Hamiltonian can be given, following proofs of the adiabatic principle (e.g., [250-252]). The last source, [252] derives an evolution operator U, which is written there, with some slight notational change, in the form... 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

The BDD approach has been applied in a number of studies that employ the parameters derived from the experimental findings [77-85]. The approach has been extended by Duxbury an co-workers, particularly Alijah in its present version, involving the new stretch-bender Hamiltonian [84,85], which follows the idea by HBl [60], it approaches the methods we tentatively call benchmark. ... 

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