Complex wavefunction

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

In this equation, T is the wavefunction, in is the mass of the particle, h is Planck s constant, and V is the potential field in which the particle is moving. The product of P with its complex conjugate ( P P, often written as P) is interpreted as the probability distribution of the particle. 

Wavefunctions are often complex quantities, and we have to be careful to distinguish a wavefunction vp from its complex conjugate P. For most of this text, wavefunctions will be real quantities and so we can drop the complex conjugate sign without lack of mathematical rigour. 

I have included the modulus bars in IV (r)p because wavefunctions can be complex quantities. For most of this and subsequent chapters, I will assume that we are dealing with real wavefunctions. 

The integration is over the coordinates of all of the electrons, and I have assumed that the wavefunction is a real quantity. In the case of a complex wavefunction we are concerned with... 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

A mistake often made by those new to the subject is to say that The Laporte rule is irrelevant for tetrahedral complexes (say) because they lack a centre of symmetry and so the concept of parity is without meaning . This is incorrect because the light operates not upon the nuclear coordninates but upon the electron coordinates which, for pure d ox p wavefunctions, for example, have well-defined parity. The lack of a molecular inversion centre allows the mixing together of pure d and p ox f) orbitals the result is the mixed parity of the orbitals and consequent non-zero transition moments. Furthermore, had the original statement been correct, we would have expected intensities of tetrahedral d-d transitions to be fully allowed, which they are not. 

Fig. 5.8 Molecular radial wavefunctions for the ferric complex FeCLt compared to the radial wavefunctions of the free ions Fe ", Fe ", and Fe " (taken from [79])...

Fig. 2 The experimentally determined potential energy V(), expressed as a wavenumber for convenience, as a function of the angle in the hydrogen-bonded complex H20- HF. The definition of

We note from Fig. 2 that the hypothetical equilibrium conformation is pyramidal, with 0e = 46(8)°, even though the geometry of the complex is effectively planar in the zero-point state (i.e. the vibrational wavefunction has C2v symmetry) because the PE barrier at the planar (0 = 0) form is low. At the time of the publication of [112] this was a critical result because it demon-... 

It is possible that the complexes benzene- -HX can be described in a similar way, but in the absence of any observed non-rigid-rotor behaviour or a vibrational satellite spectrum, it is not possible to distinguish between a strictly C6v equilibrium geometry and one of the type observed for benzene- ClF. In either case, the vibrational wavefunctions will have C6v symmetry, however. 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

Unfortunately, the many-electron wavefunction itself does not necessarily provide insight into the chemistry of complex molecules as it describes the electronic distribution over the whole system. It is therefore assumed that the true many-electron wavefunction. can be represented as the product of a series of independent, one-electron wavefunctions ... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Note that here bracket does not mean just any round, square, or curly bracket but specifically the symbols and > known as the angle brackets or chevrons. Then ( /l is called a bra and Ivp) is a ket, which is much more than a word play because a bra wavefunction is the complex conjugate of the ket wavefunction (i.e., obtained from the ket by replacing all f s by -i s), and Equation 7.6 implies that in order to obtain the energies of a static molecule we must first let the Hamiltonian work to the right on its ket wavefunction and then take the result to compute the product with the bra wavefunction to the left. In the practice of molecular spectroscopy l /) is commonly a collection, or set, of subwavefunctions l /,) whose subscript index i runs through the number n that is equal to the number of allowed static states of the molecule under study. Equation 7.6 also implies the Dirac function equality... 

Our task is now to write out the spin Hamiltonian Hs, to calculate all the energy-matrix elements in Equation 7.11 using the spin wavefunctions of Equation 7.14 and the definitions in Equations 7.15-7.17, and to diagonalize the complete E matrix to get the energies and the intensities of the transitions. We will now look at a few examples of increasing complexity to obtain energies and resonance conditions, and we defer a look at intensities to the next chapter. 

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