Nondegenerate ground states

The first Hohenberg-Kohn theorem states that, for a nondegenerate ground state, there is a one-to-one mapping among p. V. and iq... 

Fig. 4. Band diagram of a nondegenerate ground state conducting polymer.

These results, as most related results of density functional theory, have direct connections to the fundamental statement of the Hohenberg-Kohntheorem the nondegenerate ground state electron density p(r) of a molecule of n electrons in a local spin-independent external potential V, expressed in a spin-averaged form as... 

PPLB showed that for a nondegenerate ground state, if nt equals 1 for all the spin-orbitals below N0, equals 0 for all the spin-orbitals above N0, and equals t for the spin-orbital corresponding to No, then [23,26,27]... 

The energy of the nondegenerate ground state is given by the summation of the various oscillators over the characteristic frequency ... 

Our object of interest is a many electron finite system (such as an atom, molecule, cluster etc.), having, by assumption, a nondegenerate ground state (GS) (this assumption will be removed in Sects. 4.4 and 5). The numter of electrons N and the electron-nuclei potential energy v(r) = Ve (r) (the so-called external potential) are given and common for all schemes to be discussed. The GS energy qs aod the GS wave function Vqs of the system can be found from a variational principle as... 

The constraints at Eqs. (65) and (68) are important checks on the accuracy of (non-variationally derived) approximate potentials, as they are usually not fulfilled by approximate potentials except in those cases where the fulfillment of these constraint is caused for symmetry reasons such as the spherical symmetry in atoms with a nondegenerate ground state. In the case of molecules these constraints will in general not be equal to zero for non-variationally derived potentials. 

Theorem 1 The 2-RDM for the antisymmetric, nondegenerate ground state of an unspecified N-particle Hamiltonian H with two-particle interactions has a unique preimage in the set of N-ensemble representable density matrices D. 

A theorem due to Hohenberg and Kohn points to the central role of the electron density in representing the properties of a system. In 1964, Hohenberg and Kohn (1964) proved that the properties of a system with a nondegenerate ground state are unique functionals of the electron density. 

The idea of the ec methodology originated in our studies of semi-empirical cyclic polyene models Cn n with a nondegenerate ground state N —... 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

Since nearby excited states can lead to short 7ys and hence very broad absorption lines, we find for octahedral complexes that only d3, d5, and room temperatures. The other configurations must be studied at low temperatures or in complexes with symmetries that produce orbitally nondegenerate ground states. 

For an ion in a crystal field which gives a nondegenerate ground state and no nearby excited states, g can be obtained from Eq. (74). The values... 

Now consider a T = 0 system with n = 0 quanta of energy. As shown in the diagram below, there is only one possible way (ft =o = 1) to assign each molecule to its nondegenerate ground state E0 ... 

Many inorganic ions are not readily detected because they have orbitally degenerate ground states. Spin relaxation in these states is so rapid that the EPR spectrum cannot be observed at typical temperatures where electrochemical experiments are carried out. Temperatures of 4 K or lower are required to observe these ions. Table 29.1 lists transition metal ions that occur in nondegenerate ground states due to crystal field splittings and as such are amenable to observation. 

Table 29.1 3d Ions with Nondegenerate Ground States...

A change in coordination number from six to four perturbs the electronic relaxation times of nickel(II) complexes, in a sense opposite to that observed in cobalt(II) complexes. Nickel(II) complexes, when six coordinated, have an orbitally nondegenerate ground state in octahedral symmetry, with the first excited state at several thousands wave numbers in pseudotetrahedral symmetry, on the contrary, there are always closely spaced levels originating from a threefold... 

D McBranch, MB Sinclair. Ultrafast photoinduced absorption in nondegenerate ground-state conjugated polymers Signatures of excited states. In NS Sariciftci, ed. Nature of the Photoexcitations in Conjugated Polymers Semiconductor Band vs. Exciton Model, New York World Scientific, 1997. 

For nondegenerate ground state, Ivanov and Levy [26] have arrived at... 

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