Hypervirial Theorems. Development and Applications of the Hypervirial Methodology to Solve Quantum Chemistry Models

Abstract

There are a very few quantum mechanical models that can be exactly solv ed. The most important and well-known ones are perhaps the hydrogen atom and the harmonic oscillator. The Schrödinger equation for almost all the problems the theoretical chemists and physicists have to deal with cannot be solved in a closed way. Due to this, several approximate methods are currently used to obtain eigenvalues, eigenfunetions and expectation values of physical observables. The most widespread procedures are probably the perturbation theory and the variation method which are describ ed in all textbooks on quantum chemistry and quantum mechanics. Since 1960 the hypervirial theorems have become a very useful tool in handling quantum chemical and quantum mechanical problems. They are simple mathematical relationships, which are called hypervirial relationships, that the trial wavefunction should obey if it is supposed to be an acceptable approximation to the actual wavefunction.