THEOREMS OF QUANTUM MECHANICS

The Schrddinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

Before starting, we introduce some notations for the integrals we will be using. The definite integral over all space of an operator sandwiched between two functions occurs often, and various abbreviations are used  

The notations and (m A n) imply that we use the complex conjugate of the function whose letter appears first. The definite integral /ft,Af dr is called a matrix element of the operator A. Matrices are rectangular arrays of numbers and obey certain rules of combination (see Section 7.10). 

For the definite integral over all space between two functions, we write 

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A basic theorem of quantum mechanics, which will be presented here without proof, is If a and commute, namely [a, / ] = 0, there exists an ensemble of functions that are eigenfunctions of both a and - and inversely. 

For a closed system, therefore, the expectation value of the energy is constant in time, which is the energy theorem of quantum mechanics. 

There are other noteworthy single excited-state theories. Gorling developed a stationary principle for excited states in density functional theory [41]. A formalism based on the integral and differential virial theorems of quantum mechanics was proposed by Sahni and coworkers for excited state densities [42], The local scaling approach of Ludena and Kryachko has also been generalized to excited states [43]. 

The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles. Expressed somewhat loosely, the fundamental postulates of quantum mechanics assert dial microscopic systems are described by wave functions diat completely characterize all of die physical properties of the system. In particular, there aie quantum mechanical operators corresponding to each physical observable that, when applied to the wave function, allow one to predict the probability of finding the system to exhibit a particular value or range of values (scalar, vector. 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

Bader has shown that the topological partitioning of the molecules into atomic basins coincides with the requirements of formulating quantum mechanics for open systems [93], and in this way all the so-called theorems of quantum mechanics can be derived for an open system [94], Furthermore, the zero-flux condition, Eq. 1, turns out to be the necessary constraint for the application of Schwinger s principle of stationary action [95] to a part of a quantum system [93], The successful application of QTAIM to numerous chemical problems has thus deep physical roots since it is a theory which expands and generalises quantum mechanics themselves to include open and total systems, both treated on equal formal footing. 

Here, n is the number of electrons, the (pi s are the space functions, and a and are the functions associated with the electron spin. (The choice of the structure of the determinant is imposed by a general theorem of quantum mechanics, which says that the wave function must change its sign when two electrons are interchanged). We leave aside the case of molecules whose levels are not doubly occupied, viz. radicals 29). 

The Hellmann-Feynman theorem is a fundamental, yet rather simple theorem of quantum mechanics that describes the first derivatives of energy with respect to a parameter a satisfying some constraints. According to the theorem, the derivative of the energy is the expectation value of the derivative of the Hamiltonian ... 

Chapter 7 Theorems of Quantum Mechanics 7.2 HERMITIAN OPERATORS... 

Kato, T. 1950. On the adiabatic theorem of quantum mechanics. Journal of the Physical Society of Japan 5 435. 

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