Wavefunctions complex valued

Interpreting the complex energy value is simple The real part of the energy gives the position Eo of the resonance and its imaginary part the width F, by E — Eq — ij. Wavefunction related values like photoabsorption coefficient have to be independent from the complex rotation. Therefore we have to recover the correct um-otated wavefunction. In contrast to bound states there exists to any allowed real energy value E a wave function in the continuum, which can be derived from the computed, complex rotated resonance states by [5]... 

This section is devoted to the analysis and interpretation, from a physical point of view, of the LMC, FS, and CR complexity values and information planes corresponding to all neutral atoms throughout the Periodic Table, within the range of nuclear charges Z = 1-103. Such a study is carried out in position, momentum, and phase/product spaces, which corresponding distributions and their complexities are obtained by means of the accurate wavefunctions provided in Ref. [68]. 

The natural orbitals, which are eigenfunctions of the first order density matrix, are among the well-known quantities in standard quantum chemistry. However, the natural orbitals resulting from a time-dependent electronic wavefunction such as SET and PSANB are complex-valued in general, and... 

Once a complex valued electronic density matrix is in hand, it is not difficult at all to calculate the electron flux and it is not difficult either to transform the flux information to photoelectron signals (from body frame to laboratory frame), as successfully accomplished by Bandrauk et al. [499, 500], On the other hand, it is not as simple to track the simultaneous dynamics of remaining electrons in the molecular site while photoelectrons are leaving away. Note that even if we successfully have a rescaled density p (r, t) in equation (7.2) with an estimate of the flux, we can make infinitely many (untrue) electronic wavefunctions that can reproduce this same p r,t). The key to be considered is, therefore, to assign which electronic configurations the photoelectrons are leaving from and how consistently we can describe the states of the remaining electronic wavepacket. To do so, we think it to be most appropriate to use natural orbital representation of the electron flux. 

The outline of the procedure is as follows. (1) Run an electron wavepacket dynamics at a certain time, which gives rise to a complex valued electronic wavefunction. (2) Calculate a set of complex-valued natural orbitals (NO) with real-valued occupation numbers. (3) With these NOs, estimate the outgoing electron flux at a predetermined sm-face area of the molecule and... 

If we worked only with real functions, our phase factors would be 1 or — 1. However, wavefunctions can have imaginary components, and so can the phase factors. A general way to represent the phase that allows for complex values is by a factor e , where cp varies from 0 to 2ir. The Euler formula (see Problem A.9) allows us to write this as... 

Actually, this derivation of Eq. 2.17 includes one other assumption, which was not mentioned because it doesn t affect K. We used the unnormalized function sin(/cx + 0) as the general form of a wavefunction with well-defined kinetic energy. There s a hidden assumption here that the wavefunction has no complex values, that it is a pure real function. 

The square of this function will result in a complex value. To ensure that the probability density has a real value, the probability density is obtained by multiplying the wavefunction by the complex conjugate of the wavefunction. The complex conjugate is obtained by replacing any i in the function with a -i . The complex conjugate of the function above is... 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

Definition (5) shows that TJb which is sometimes called the electronic matrix element , represents the residual interaction resulting from the overlap of the wavefunctions v /j and These functions, which describe the initial and final electronic states of the whole system, respectively, depend closely on the nature of the redox centers and of the medium, so that reliable values of T are very difiicult to obtain from ab initio calculations in complex systems. For that reason, some authors have proposed determining T b semi-empirically by using the results of spectroscopic measurements. We begin by a brief presentation of... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

This picture can qualitatively account for the g tensor anisotropy of nitrosyl complexes in which g = 2.08, gy = 2.01, and g == 2.00. However, gy is often less than 2 and is as small as 1.95 in proteins such as horseradish peroxidase. To explain the reduction in g from the free electron value along the y axis, it is necessary to postulate delocalization of the electron over the molecule. This can best be done by a complete molecular orbital description, but it is instructive to consider the formation of bonding and antibonding orbitals with dy character from the metal orbital and a p orbital from the nitrogen. The filled orbital would then contribute positively to the g value while admixture of the empty orbital would decrease the g value. Thus, the value of gy could be quite variable. The delocalization of the electron into ligand orbitals reduces the occupancy of the metal d/ orbital. This effectively reduces the coefficients of the wavefunction components which account for the g tensor anisotropy hence, the anisotropy is an order of magnitude less than might be expected for a pure ionic d complex in which the unpaired electron resides in the orbital. 

Such fluctuations of the photon flux, emitted from a molecule, have been predicted to be due to cooperative effects (21.22). The theory is based on an idea of Prigogine and coworkers (s. e.g.(22)) who treated the irreversible part of a physical process by transforming the wavefunctions of a dissipative system into another space using a "dynamical" non-unitary representation D = exp(-iVT /fI) with a "star-Hermitian" time operator 3 and V describing the interaction of a relevant local system Hq, e.g. the complex chromophore, and the total system H, i.e. our crystal. In the new representation y>=D Y no additional time dependence is introduced, dD/dt = 0, any expectation value of an operator M=DMD should be unchanged = M> and the total Hamiltonian is transformed by 1T=DHD 1 = Hq to the local system Hamiltonian (21.22). To describe the time development in the new representation, the electron density... 

The electronic matrix element, ( P/ /7 P,), couples the electronic wavefunctions of the precursor complex, Fj, and the successor complex, M, through the exchange operator //. When the reaction is adiabatic, the value of the factor k measuring the... 

The probability density P(x) = f(x) 2 is the same for f as it is for —f the expectation values for all observable operators are the same as well. In fact, we can even multiply f by a complex number and the same result holds. The overall phase of the wavefunction is arbitrary, in the same sense that the zero of potential energy is arbitrary. Phase differences at different points in the wavefunction, on the other hand, have very important consequences as we will discuss shortly. 

This Report deals with the calculation of spectroscopic constants both for diatomic and for polyatomic molecules, while concentrating on the former. The spectroscopic constants included are restricted to those measured in high-resolution gas-phase work. We cover the whole range of complexity in computation since this is determined not by the method used in computing the expectation value, but by the quality of the wavefunction used. Generally the wavefunctions used are of the ab initio type, but their quality will depend on the size and type of basis set employed as well as the method. 

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