Wavefunction meaning

A physical interpretation of the exchange energy associated with the Pauli principle can be advanced (see, for example, ref. 45). The spin correlation introduced by the antisymmetrization of the wavefunction means that, independently of the Coulomb repulsion, the probability of two electrons being close in space is smaller for electrons with the same spin than for... 

This is not because these terms do not contribute to the change in any particular occupied orbital but because, as we noted above, the single-determinant nature of the total wavefunction means that all these contributions will cancel giving no net change to the total wavefunction. So the correction to each occupied orbital coefficient vector may be written ... 

Figure 5.4 gives a pictorial representation of the way in which the three sp hybrid orbitals are constructed. Remember that a change in sign for the atomic wavefunction means a change in phase. The resultant directions of the lower two hybrid orbitals in Figure 5.4 are determined by resolving the vectors associated with the 2p and 2py atomic orbitals. 

The spectrum in fig. 8 shows that there is strong mixing between the Ce4f and the O 2p wavefunctions, meaning a considerable covalent bond and therefore the simple valence term, which is so convenient to describe most lanthanide compounds (with the exception of the mixed valent systems), loses some of its simplicity. This is emphasized by the band structur calculation of Koelling et al. (1983), which finds just that hybridization. However the question of the size of the 4f population, and therefore of the valence, as determined by these calculations, is also one of the choice of the spheres for the integration >f the charges around the ions. 

In the absence of external fields, these 36 states occur in 10 energy levels, one for each term. These lie at different energies for several reasons. We have already seen, in our discussion of ls2s helium stales, that different spin multiplicities are associated with different symmetries of the spin wavefunction, meaning that the space part of the wavefunctions also differ in symmetry. This has a significant effect on energy, so... 

Hamiltonian = t+ The additivity of implies that the mean-field energies il/are additive and the wavefunctions [Pg.2162]

Because of the indistingiiishability of the electrons, the antisynnnetric component of any such orbital product must be fonned to obtain the proper mean-field wavefunction. To do so, one applies the so-called antisynnnetrizer operator [24] A= Y.p -lf p, where the pemuitation operator mns over all A pemuitations of the N electrons. Application of 4 to a product fiinction does not alter the occupancy of the fiinctions ( ). ] in it simply scrambles the order which the electrons occupy the ( ). ] and it causes the resultant fiinction... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

The function/( C) may have a very simple form, as is the case for the calculation of the molecular weight from the relative atomic masses. In most cases, however,/( Cj will be very complicated when it comes to describe the structure by quantum mechanical means and the property may be derived directly from the wavefunction for example, the dipole moment may be obtained by applying the dipole operator. 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

Before moving deeper into understanding what quantum mechanics means, it is useful to learn how the wavefunctions E are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems. Knowing the solutions to these easy yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics because these model cases can be used as concrete examples. ... 

Assume that an experiment has been carried out on an atom to measure its total angular momentum L. According to quantum mechanics, only values equal to L(L+1) h will be observed. Further assume, for the particular experimental sample subjected to observation, that values of equal to 2 and 04f were detected in relative amounts of 64 % and 36%, respectively. This means that the atom s original wavefunction / could be represented as ... 

It should be emphasized that the proeess of deleting or erossing off entries in various Ml, Ms boxes involves only counting how many states there are by no means do we identify the particular L,S,Ml,Ms wavefunctions when we cross out any particular entry in a box. For example, when the piapoP product is deleted from the Ml= 1, Ms=0 box in accounting for the states in the level, we do not claim that piapoP itself is a member of the level the poOtpiP product state could just as well been eliminated when accounting for the P states. As will be shown later, the P state with Ml= 1, Ms=0 will be a combination of piapoP and pootpiP. ... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction. 

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition ... 

Electrons are indistinguishable, they simply cannot be labelled. This means that an acceptable electronic wavefunction has to treat all electrons on an equal footing. Thus, although 1 have so far implied that electron 1 is to be associated with nucleus Ha, and electron 2 with nucleus Hb, 1 must also cater for the alternative description where electron 1 is associated with nucleus Hb and electron 2 with nucleus Ha. 1 therefore have to modify Table 4.1 to Table 4.2. 

I don t mean that such a wavefunction is necessarily very accurate you saw a minute ago that the LCAO treatment of dihydrogen is rather poor. I mean that, in principle, a Slater determinant has the correct spatial and spin symmetry to represent an electronic state. It very often happens that we have to take combinations of Slater determinants in order to make progress for example, the first excited states of dihydrogen caimot be represented adequately by a single Slater determinant such as... 

What does this equation tell us For this wavefunction, the angular wavefunction Y is a constant, 1/2tti/2 , independent of the angles, which means that the wavefunction is the same in all directions. The radial wavefunction R(r) decays exponentially toward zero as r increases, which means that the electron density is highest close to the nucleus (e° =1). The Bohr radius tells us how sharply the wavefunction falls away with distance when r = a ), t i has fallen to 1/e (37%) of its value at the nucleus. 

Cjl.115 Wavefunctions are normalized to 1. This term means that the total probability of finding an electron in the system is 1. Verify this statement for a particle-in-the-box wavefunction (Eq. 10). 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

Again the left superscript indicates the spin-triplet nature of the arrangement. The letter A means that it is spatially (orbitally) one-fold degenerate and it is upper-case because we describe two-electron wavefunctions. The subscript is g because the product of d orbitals is even under the octahedral centre of inversion, and the right subscript 2 must remain a mystery for us once again. 

Note that, throughout this discussion, we have used lower-case letters when refering to orbitals and upper-case when we mean many-electron wavefunctions. There arises the question of, what are the relationships between I and L, or between s and S . They are determined by the vector coupling rule. This states that the angular momentum for a coupled (i.e. interacting) pair of electrons may take values ranging from their sum to their difference (Eq. 3.11). 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

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