Eigenfunctions table

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

The formation of a stable coordination compound involving the four tetrahedral spz eigenfunctions might decrease the L contribution appreciably. It was indeed pointed out by Bose that in the compounds listed in the last column of Table II the observed moments approach more closely the theoretical values yu5. 

If the bonds are ionic or ion-dipole bonds, the magnetic moments are those of the isolated central ions, given in the first column of moments in Table III. If the complex involves electron-pair bonds formed from sp alone, such as four tetrahedral sp3 bonds, the magnetic moments are the same, for the five d eigenfunctions are still available for the remaining electrons. The hydrazine and ammonia complexes mentioned above come in this class. 

The observed diamagnetism of the ions [Mo(CN)s]4- and [W(CN)8]4 shows that the central atom forms eight electron-pair bonds, involving the eigenfunctions dispi (fourth column of Table III). 

The energies before and after the interaction are shown in Fig. 5. The eigenfunctions and eigenvalues are listed in Table 1. 

Table 1. Eigenfunctions and eigenvalues of the interaction problem involving do and the four determinants which can be formed from an i - T orbital promotion...

The dimensions of the spin spaces for the active electrons in Table 2, cf. Eq. (9)) are certainly not small. It proved difficult to find a spin basis in which very few of the coefficients were large and so we adopted instead a spin correlation scheme cf. Section 4.2). In the present work, we exploited the way in which expectation values of the two-electron spin operator evaluated over the total spin eigenfunction 4, depend on the coupling of the individual spins associated with orbitals ( )/ and j. Negative values indicate singlet character and positive values triplet character. Special cases of the expectation value are ... 

Problem 8-12. Verify that if -ip is an eigenfunction of H with eigenvalue , then the expectation value of the energy is equal to . The expression (/, g) is called the inner product of fund g. It has a number of properties analogous to those of the dot product of two vectors. These are illustrated in Table 8.2. 

Thus, a and fi are eigenfunctions of the operator S, with eigenvalues of 1/2 and —1/2, respectively, in atomic units (recall that the value of h is 1 in atomic units, see Table 1.1). The spin operator. S is defined by... 

Table 3.2. Angular momentum eigenfunctions in LS-coupling for the electron configuration 2p2. From J. C. Slater, Quantum theory of atomic structure (1960), with the kind permission of J. F. Slater and The McGraw-Hill Companies.

Molecular orbitals (MOs) were constructed using linear combinations of basis functions of atomic orbitals. The MO eigenfunctions were obtained by solving the Schrodinger equations in numerical form, including Is— (n+l)p, that is to say, Is, 2s, 2p, -ns, np, nd, (n+l)s, (n+l)p orbitals for elements from n-th row in the periodic table and ls-2p orbitals for O, where n—1 corresponded to the principal quantum number of the valence shell. 

Table 3.1 Eigenfunctions Harmonic Oscillator H = — i//Jx) of the Hamiltonian Operator for the h2/2m) d2/dx2) + knX2...

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