Separable wavefunction

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

Consider the RDMs obtained from the separable wavefunction in Eq. (12). Since a and b are strongly orthogonal, it follows from Eq. (8) that ( a b a flj I a b) = 0 unless 0, and (f)j are associated with the same subsystem. Thus the 1-RDM separates into subsystem 1-RDMs,... 

In numerous applications, however, we have found it sufficiently accurate, at least for the vibrational ground state of the parent molecule, to assume a separable wavefunction directly in terms of Jacobi coordinates,... 

The electron density (10) is the so-called diagonal element of a more general quantity, the (spinless) one-electron density matrix, P(r, r ), defined in exactly the same way except that the variables in it carry primes - which are removed before the integrations. The reduction to (11), in terms of a basis set, remains valid, with a prime added to the variable in the starred function. For a separable wavefunction, the density matrices for the whole system may be expressed in terms of those for the separate electron groups in particular, for a core-valence separation,... 

Fock recognized that the separable wavefunction employed by Hartree (Eq. (1.6)) does not satisfy the Pauli exclusion principle. Instead, Fock suggested using the Slater determinant... 

Wc have taken the initial coordinates and momenta for the swarm of trajectories from the Wigner distribution. It can be easily proven that for a separable wavefunction the Wigner function can be written in a product form. As a basis for photodissociation calculations we have used a wavefunction in the form 3, therefore, the corresponding Wigner distribution is given by an expression... 

The delightful thing about one-electron operators is that we can exactly solve the Schrodinger equation if the Hamiltonian is approximated by its one-electron part V = S, h(i) = Hq) since a separable wavefunction can be construaed as a produa of one-particle funaions, ,(ry). 

The total wavefunction is obtained by joining together the separate wavefunctions from the three regions. Ideally, this should be done in such a way that there is no discontinuity in either i/r or dy dx at the boundaries between the regions. Thus, at the boundary between regions 1 and 2, we have yr, = y/ and dy/ /dx = dy/Jdx, and similar conditions apply at the boundary between regions 2 and 3. An example of how the real parts of the wavefunctions can be joined up is shown in Figure 4.8. 

Despite the separated wavefunctions in one dimension each, it is important to understand that the operator must operate on the entire wavefunction. Although the entire wavefunction is in three dimensions, the one-dimensional operator acts only on the part that depends on the coordinate of interest. 

NEDA evaluates AE by first performing separate wavefunction calculations on each monomer A, B (in its geometry in the complex) with the fidl dimer basis set, corresponding to the counterpoise-corrected binding energy (as defined by S. F. Boys and F. Bemardi, Mol. Phys. 19, 553, 1970), namely,... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

For separable initial states the single excitation terms can be set to zero at all times at this level of approximation. Eqs. (32),(33),(34) together with the CSP equations and with the ansatz (31) for the total wavefunction are the working equations for the approach. This form, without further extension, is valid only for short time-domains (typically, a few picoseconds at most). For large times, higher correlations, i.e. interactions between different singly and doubly excited states must be included. 

The semiclassical approach to QCMD, as introduced in [10], derives the QCMD equations within two steps. First, a separation step makes a tensor ansatz for the full wavefunction separating the coordinates x and q ... 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

We need to investigate the conditions under which this is true, and to do this we make use of a technique called separation of variables . We substitute the product wavefunction (3.3) into (3.2) to give... 

In order to investigate whether the wavefunction can indeed be written in this way, we use the separation of variables technique and so write a wavefunction of the form... 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

Regarding the emission properties, AM I/Cl calculations, performed on a cluster containing three stilbene molecules separated by 4 A, show that the main lattice deformations take place on the central unit in the lowest excited state. It is therefore reasonable to assume that the wavefunction of the relaxed electron-hole pair extends at most over three interacting chains. The results further demonstrate that the weak coupling calculated between the ground state and the lowest excited state evolves in a way veiy similar to that reported for cofacial dimers. 

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