Wave function probability

The Meaning of the Wave Function Probability and Electron Density 57... 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

Fig. 10.1. Absence of electronic correlation in the helium atom as seen by the Hartree-Fock method. Visualization of the cross-section of the square of the wave function (probability density distribution) describing electron 2 within the plane xy provided electron 1 is located in a certain point in space a) at (—1,0,0) b) at (1,0,0). Note, that in both cases the conditional probability density distributions of electron 2 are identical. This means electron 2 does not react to the motion of electron 1, i.e. there is no correlation whatsoever of the electronic motions (when the total wave function is the Hartree-Fock one).

Marcus R A 1970 Extension of the WKB method to wave functions and transition probability amplitudes (S-matrix) for inelastic or reactive collisions Chem. Phys. Lett. 7 525-32... 

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles... 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

Basis sets can be further improved by adding new functions, provided that the new functions represent some element of the physics of the actual wave function. Chemical bonds are not centered exactly on nuclei, so polarized functions are added to the basis set leading to an improved basis denoted p, d, or f in such sets as 6-31G(d), etc. Electrons do not have a very high probability density far from the nuclei in a molecule, but the little probability that they do have is important in chemical bonding, hence dijfuse functions, denoted - - as in 6-311 - - G(d), are added in some very high-level basis sets. 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

Chemists are able to do research much more efficiently if they have a model for understanding chemistry. Population analysis is a mathematical way of partitioning a wave function or electron density into charges on the nuclei, bond orders, and other related information. These are probably the most widely used results that are not experimentally observable. 

FIGURE 1 3 Boundary surfaces of the 2p orbitals The wave function changes sign at the nucleus The two halves of each orbital are indicated by different colors The yz plane is a nodal surface for the Ip orbital The probability of finding a electron in the yz plane is zero Anal ogously the xz plane is a nodal surface for the 2py orbital and the xy plane is a nodal surface for the 2pz orbital You may examine different presentations of a 2p orbital on Learning By Modeling... 

Nodal surface (Section 1 1) A plane drawn through an orbital where the algebraic sign of a wave function changes The probability of finding an electron at a node is zero... 

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

Figure 1.7 Plots of (a) the radial wave function (b) the radial probability distribution function and (c) the radial charge density function 4nr Rl( against p...

Figure 6.6 Wave functions (dashed lines) and probability density functions (solid...

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