Ground-state wave function hydrogen molecule

Compute S0(-Re)/S0(0) for the hydrogen-molecule ground electronic state, where S0 is the ground-state wave function of a onedimensional harmonic oscillator. 

Calculations by the self-consistent field LCAO-MO method for the ground state wave function of the pyrazine molecule indicate that the lone pairs are quite different. The lower lone pair is little delocalized (1.88 electrons on nitrogen), but the second lone pair is as delocalized as the lone pair in pyridine with 1.37 electrons on nitrogen, 0.22 electrons on hydrogen, and 0.40 electrons on carbon.63... 

Now let us look at the coupled fermions. Fillaux considers, as an example, the hydrogen molecule and argues that a similar pattern should take place for the oscillators in KHCO3. The vibrational ground-state wave functions are... 

The ground-state wave function, and the energy eigenvalue, g, of this system are expressed by Eq. (1.149) in a way analogous to the valence bond method described for the hydrogen molecule, where k is the force constant of the nucleus motion. 

Thus the wave function for an electron in the presence of the well looks like Fig. 5.6 it has the same form as a one-dimensional cross-section of the wave function for the ground state of the hydrogen atom. This correspondence suggests using two such wells to make a one-dimensional model for examining the ground state of the hydrogen-molecule ion. 

They consider two forms of an approximate wave function for the ground state of the hydrogen molecule which depend on a single parameter. These wave functions correspond to the vb and mo wave function for a particular choice of the parameter. The first wave function considered by Coulson and Fischer is written as... 

We are interested in properties of the ammonia molecule in its ground and excited states e.g.. we would like to know the mean value of the nitrogen-hydrogen distance. Only quantum mechanics gives a method for calculation this value (p. 26) we have to calculate the mean value of an operator with the ground-state wave function. But where could this function be taken from Could it be a solution of the Schrddinger equation Impossible unfortunately, this equation is too difficult to solve (14 particles cf. problems with exact solutions in Chapter 4). 

For two-electron systems, the number of Slater determinants in the FCI wave function (5.3.2) increases only as M, where M is the number of spin orbitals. For the hydrogen molecule, it is therefore possible to carry out FCI calculations in large basis sets. We here employ the cc-pV( basis, which contains a total of 60 AOs and is capable of providing an accurate description of the X ground state of the hydrogen molecule, fri this basis, there are 552 Slater determinants consistent with symmetry in the FCI wave function. 

The simplest molecular system exhibiting effects of electron correlation is the hydrogen molecule. For this molecule the explicitly correlated wave function has been applied in the early days of quantum mechanics (James and Coolidge, 1933), It was later generalised by Kolos and Wolniewicz (Kcrfos and Wolniewicz, 1965) and successfully used to solve variety of problems in the ground and excited states of the hydrogen molecule. This wave function, called there the Kolos-Wolniewicz function (Kolos and Wolniewicz, 1965) is assumed in the form of an expansion ... 

In the quantum-mechanical description of atoms and molecules, electrons have characteristics of waves as well as particles. In the familiar case of the hydrogen atom, the orbitals Is, 2s, 2p,... describe the different possible standing wave patterns of electron distribution, for a single electron moving in the potential field of a proton. The motion of the electrons in any atom or molecule is described as fully as possibly by a set of wave functions associated with the ground and excited states. 

At the liquid helium temperature, we can safely assume that all para-hydrogen molecules are populated in the t = 0, /=0 ground state. The characteristic of this ground-state para-hydrogen molecule is that it has an isotropic distribution of its molecular wave function. Therefore, the electrostatic intermolecular interaction... 

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