Bom interpretation

The electron density p(r) at a point r can be calculated from the Bom interpretation of the u a Vv tu iiclion as a sum of squares of the spin orbitals at the point r for all occupied molecular orhital-i. For a system of N electrons occupying N/2 real orbitals, we can write ... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

FIGURE 1.24 The Bom interpretation of the wavefunction. The probability density (the blue line) is given by the square of the wavefunction and depicted by the density of shading in the band beneath. Note that the probability density is zero at a node. A node is a point where the wavefunction (the orange line) passes through zero, not merely approaches zero. 

Suppose we wish to measure the position of a particle whose wave function is W(jc, i). The Bom interpretation of F(x, as the probability density for finding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state /) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value x) of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value. 

Even worse is the confusion regarding the wavefunction itself. The Bom interpretation of quantum mechanics tells us that T/r (r) (r)dr represents the probability of finding the particle with spatial coordinates r, described by the wavefunction i//(r), in volume element dr. Probabilities are real numbers, and so the dimensions of jr(r) must be of (length)-3 2. In the atomic system of units, we take the unit of wavefunction to be ao 3/2-... 

Consider first the situation where a particle moves in two dimensions, labelled x and y. The wavefunction now depends on these two variables. As before, the Bom interpretation shows that its square gives the probability of finding the particle at some position (x,y). The two-dimensional form of Schrodingef s equation is... 

P(r) gives the probability that an electron in the corresponding orbital will be found at a distance r from the nucleus. More strictly, P(r)dr equals the probability of finding the electron somewhere in the spherical shell between radii r and r + dr. According to the Bom interpretation, y/1 gives the... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

The Bom interpretation of quantum mechanics tells us that (r, s) dr ds gives the chance of finding the electron in the spatial volume element dr and with spin coordinate between s and s + ds. Since probabilities have to sum to 1, we have... 

Ball and stick representation Basis function 114 BLYP 314 Bohr orbit 22 Bohr radius 19 Bohr theory 1 Boltzmann s law 61 Bond orbital 129 Bond order 126 Bond separation reaction 319 Bond-length alternation 126 Bom interpretation 23, 32, 100 Bom-Oppenheimer approximation 86, 230, 265, 273... 

Consider the interaction of two electrons, e and c2, that are located in the AOs and (frv. We do not exclude the possibility that the two electrons are in the same AO, pi = v, provided that they have opposite spin. The time-averaged distribution of electron 1 is given by 2(ci)dvi and that of electron 2 by 2(c2)dv2 (Bom interpretation). Therefore, the... 

According to the Bom interpretation of the wavefunction, lelectron density of electron 1 in orbital IcTg at a position ri. Similarly, lag(r2)lcTg(r2) is the electron density of electron 2. The electrostatic repulsion between these regions of electron density thus equals lcTg(ri)lag(ri) x (l/ri2) x lag(t2)l(Tg(t2), where rx2 is the distance between the two electrons. The integral of this function over aU space thus corresponds to the electrostatic (Coulomb) repulsion between the two orbitals. 

We have already encountered the particle density distribution in the context of the Bom interpretation in section 4.1. After what has been said about observables, it should be possible to assign an operator to this observable also. We can deduce the explicit form of the operator pr> which represents the particle density at a given position r in space, by relating its expectation value for an N-particle system,... 

The Bom interpretation leads to a number of important implications on the wavefunction. The function must be single-valued it would not make physical sense that the particle had two different probabilities in the same region of space. The sum of the probabilities of finding a particle within each segment of space in the universe ( /n Vn times a volume element, dx) must be equal to unity. The mathematical operation of ensuring that the sum overall space results in unity is referred to as normalizing the wavefunction. 

The wavefunction for Region II contains an exponentially increasing component as x increases. As x approaches infinity, the wavefunction in Region II will approach infinity. This is an untenable result based on the Bom interpretation - the probability density of the particle will approach infinity as wavefunction approaches infinity. Since this is not physically possible, the positive exponential component of the wavefunction must be discarded. 

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