Quantum mechanical observable

As with atomic charges, the bond order is not a quantum mechanical observable and so anuus methods have been proposed for calculating the bond orders in a molecule. 

This exercise will examine other ways of computing charges other than Mulliken population analysis. Since atomic charge is not a quantum mechanical observable, all methods for computing it are necessarily arbitrary. We ll explore the relative merits of various schemes for partitioning the electron density among the atoms in a molecular system. 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

This section is based mainly on work published by Miller and Schafer17 and Hickman and Morgner.22 As in the preceding section, the intention is to show how the electronic quantities V0(R), F+(/ ), and T(R) can be used to calculate—this time quantum mechanically—observable quantities. 

From a chemical standpoint, the decomposition of the energies of the double bonds into the corresponding a and tz components, Ea and En, is important. Unfortunately, this is a conceptual division and these quantities are not quantum-mechanical observables. Thus, approximate methods have been devised to estimate these contributions. 

In the formalism of quantum mechanics, observables are associated to hermitian operators that act on the Hilbert space of square integrable functions representing the state of the quantum system. In the following, for the sake of definiteness, we shall consider hermitian operators B which can be written as hermitain combinations of position and momentum operators,... 

Consider two quantum-mechanical observables A and B with thermal equilibrium correlation functions verifying the Kubo-Martin-Schwinger condition [35], that is,... 

Molecular geometries measured with condensed-phase techniques such as X-ray diffraction or NMR cannot be regarded as inherent properties of isolated species. Similarly, as the "determination" of molecular geometries from microwave spectra involves collation of data pertaining to many spectroscopic states of species differing in isotopic compositions, such geometries are merely collections of fitting parameters that cannot be viewed as quantum-mechanical observables. 

One set of quantities often evaluated from the ground-state wave function that are not quantum-mechanical observables are the various components of the Mulliken population analysis (Mulliken, 1955, 1962). For example, we could define the net Mulliken charge on an atom A as ... 

There has been considerable effort directed towards the assessment of the covalency or ionicity of various solids. The output of standard ab initio (SCF) Hartree-Fock-Roothaan calculations contains Mulliken (1955) charge distribution analysis parameters such as the atomic-orbital populations, net atomic charges, and bond overlap populations deseribed earlier (Chapter 3), which are often used to discuss the relative covalency or ionicity of materials. Considerable caution is required in using such parameters, however, since net atomic charges and other such quantities are not quantum-mechanical observables that is, they cannot even in principle be measured, and are highly basis-set dependent, as noted by Hehre et al. (1986 pp. 336-41). This is illustrated for molecules more relevant to mineralogy in Table 7.1, in which a number of properties of CO2 and SiOj are shown calculated at various basis-set levels. It is clear... 

I. Every observable quantity can be calculated, at least in principle, from the density alone, i.e. each quantum mechanical observable can be written as a functional of the density,... 

The general principles of entanglement are valid for all quantum-mechanical observables, including momenlum. spin and so un. In practice, however, the mo.st accessible entanglement pheiioiiiena involve photon polari/.ations. This is true... 

Because atomic charge is not a quantum mechanical observable, we must use some indirect method to calculate these values. Moreover, since we lack experimental results to guide us, other methods of validating our assignments must be devised. Accurate reproduction of some observable, whether experimentally measured (dipole moment) or determined directly from the wavefunction (electrostatic potential, molecular moments, etc.), increases our confidence in the reliability of the assigned charges. 

In the spirit of the opening quote of this chapter, the quantum theory of atoms in molecules (QTAIM) [63] has been extensively applied to classify and understand bonding interactions in terms of a quantum mechanical observable the electron density p(r). In this chapter we will take advantage of this theory to... 

In this section, the basic working equations of molecular QED have been given which enable fhe perturbative solution to be obtained for the quantum mechanical observable quantity for any specfroscopic or intermolecular process. Before going on to apply the formalism presented to the computation of the optical binding energy in Section 4, the QED calculation of the retarded dispersion potential is briefly discussed in the following section. 

Note that another, nontrivial advantage of the atomic charge defined by Eq. [50] is that they are given in terms of a well-defined quantum mechanical observable. This is desirable and avoids the need to parameterize the charges by fitting some other observable, such as the electrostatic potential at a set of external points. 

B. With this in mind, let us consider the two water molecules in Figure 19 at distances such that the Lennard-Jones repulsive term can be neglected and the interaction is purely electrostatic. The electrostatic energy, up to (molecular) dipole-quadrupole interactions, is given in terms of well-defined quantum mechanical observables by... 

The electrostatic potential is a quantum mechanical observable that can be obtained directly from the wavefunction. Assuming an SCF wavefunction, the electrostatic potential V(r) is defined by Eq. [52], where Z, is the atomic charge of atom k centered at R. ... 

In quantum mechanics observables are represented by linear Hermitian operators A, which change in time according to the equation of motion... 

The LevinthaTs paradox is an open problem still. To avoid the core of the problem — it s kinema-tical aspect — we propose a new approach in this regard. Actually, we treat the macromolecules conformations as the quantum-mechanical observable. Bearing in mind the foundations of the decoherence theory, we are able to model both, existence and maintenance of the conformations as well as the conformational transitions in the rather short time intervals. Our model is rather qualitative yet a general one — while completely removing the LevinthaTs paradox — in contradistinction with the (semi-)classical approach to the issue. 

As the dipole moment for polyatomic molecules can vanish due to conflicting factors, we also monitor the bond polarity by observing the Mulliken atomic populations on F and H. We recognize that unlike the dipole moment, such net populations are not quantum mechanical observables, and that the... 

---