Atomic wave functions and

In this equation, f is the molecular wave function, is an atomic wave function, and a is a weighting coefficient that gives the relative weight in the "mix" of the atomic wave functions. The summation is... 

The two bonding 7r orbitals represented by these wave functions are degenerate. The wave functions for the antibonding states are identical in form except that negative signs are used in the combination of atomic wave functions and in the normalization constants. 

Herman and Skillman [79] used an HFS algorithm to calculate radial atomic wave functions and energy eigenvalues for all atoms, tabulating all results and the computer software at the same time. They treated all single electronic... 

Now since the atomic wave functions and were previously normalized, f th and /t udr each equal one. Hence... 

If the metal and donor atom wave functions and their separations are the same, the group radial integrals may be unchanged and we can note that the result for the difference between the t2 and e antibonding rf-orbital sets, Et — E ... 

We are interested in finding the values of cx and c2 which, for given atomic wave functions and constant R (that is, for constant HAA, Hgg, and Hab ), will minimize the orbital energy W. This is of interest because it can be shown that the minimized solution W always is larger than or equal to W0, where W0 is the correct solution to the problem. Thus, the smaller W is the better. To determine cx and c2 we differentiate W with respect to these parameters. 

Fassaert et al. (68) simulated H adsorption on a Cu surface by adding an additional electron per metal atom to the system. This approximation relies on the fact that atomic wave functions and energy levels are not too different for Ni and Cu and that their principal difference lies in the number of valence electrons. In the case of adsorption to Cu substrate, which has no unfilled d orbitals, the metal d orbitals do not participate in the bonding to H. All bonding takes place using the metal 4s orbitals. The calculated covalent bond energy is comparable on the Ni and Cu substrate models, so that from the results a distinction between the catalytic properties of the two metals cannot be made. 

Under such conditions, the probability of a transition from the initial ground state of the parent ion to each electronic state of the daughter atom can be simply evaluated from atomic wave functions, and the calculations can follow the lines outlined in the preceding Section. 

Each of the above expressions for E s, E2s and E2v gives the exact energy result when the impenetrable wall is situated at the appropriate node of an unconfined atom wave function, and the error increases as the wall moves away from the node. 

The unperturbed system is two hydrogen atoms. We have two zeroth-order functions consisting of products of hydrogen-atom wave functions, and these belong to a degenerate level. The correct ground-state zeroth-order function is the linear combination (13.101). 

For Eb > Ea the term a>ba = (Eb — Ea)lh is positive. In the transition Ea Eb, the atomic system absorbs energy from the radiation field. Noticeable absorption occurs, however, only if the field frequency a> is close to the eigenfrequency (i>ba- In the optical frequency range this implies that o)ba — < < ( ba The second term in (2.70b) is then small compared to the first one and may be neglected. This is called the rotating-wave approximation for only that term is kept in which the atomic wave functions and the field waves with the phasors exp(—icu / t) = txpi+icobJ) and exp(— cot) rotate together. 

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