Lowest-energy eigenfunction

The dominant error term is third order in At. The initial wavefunction (Qx,Qy,t) at t = 0 is normally the lowest energy eigenfunction of the initial state of the spectroscopic transition. The value of the wavefunction at incremental time intervals At is calculated by using Eq. (7) for each point on the (Qx,Qy) grid. The autocorrelation function is then calculated at each time interval and the resulting < (t> is Fourier transformed according to Eq. (2) to give the emission spectrum. 

The DMC method achieves the lowest-energy eigenfunction by employing the quantum mechanical evolution operator in imaginary time [25], For an initial function expanded in eigenstates, one finds that contributions of the excited states decay exponentially fast with respect to the ground state. 

Which one of the following is the correct formula for the lowest-energy eigenfunction for a particle in a one-dimensional hox having infinite barriers at x = — Z,/2 andZ,/2 ... 

We will now discuss the lowest-energy eigenfunction of Eq. (4-6) in some detail, since an understanding of atomic wavefunctions is crucial in quantum chemistry. The derivation of formulas for this and other wavefunctions will be discussed in later sections, but it is not necessary to labor through the mathematical details of the exact solution of Eq. (4-6) to be able to understand most of the essential features of the eigenfunctions. The formula for the normalized, lowest-energy solution of Eq. (4-6) is... 

Figure 4-4 The volume-weighted probability density for the lowest-energy eigenfunction of the hydrogenlike ion. The most probable value of r occurs at rmp-...

Notice that the lowest-energy eigenfunction is finite at r = 0 even though V is infinite there. This is allowed by our arguments in Chapter 2 because the infinity in V occurs at only one point, so it can be cancelled by a discontinuity in the derivative of This is possible only if has a corner or cusp at r = 0 (see Fig. 4-3a and e). 

The variation method is based on the idea that, by varying a function to give the lowest average energy, we tend to maximize the amount of the lowest-energy eigenfunction 1/ 0 present in the linear combination already discussed. Thus, if we minimize... 

We have already seen (Chapter 4) that the lowest-energy eigenfunction for the hydrogen atom is (in atomic units)... 

DMC achieves the lowest-energy eigenfunction by employing the propagator... 

The reference propagator is calculated numerically in terms of the M lowest energy eigenfunctions 4>k and eigenvalues Ek of Ho ... 

Recall that the minimum requirement for a many-electron wave function is that it be written as a suitably antisymmetrized sum of products of one-electron wave functions, that is, as a Slater determinant of MOs [see equation (A.68)] In Chapter 2 and Appendix A, we find that the condition that this be the best possible wave function of this form is that the MOs be eigenfunctions of a one-electron operator, the Fock operator [recall equation (A.42)], from which one can choose the appropriate number of the lowest energy. The Fock operator in restricted form, F( 1) [RHF, the UHF form was given in equation (A.41)], is given by... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

A wave function of an electron configuration is generally approximated as a product of molecular orbital functions, which are eigenfunctions of a one-electron Hamilton operator. When 2n electrons occupy n molecular orbitals, the wave function of electron configuration of the lowest energy is written as... 

The lowest-energy solutions deviating from spherieal symmetry are the 2p-orbitals. Using Eqs (7.44), (7.45) and the f = 1 spherical harmonics, we find three degenerate eigenfunctions ... 

The various energy levels of the system exhibit a tunnel splitting and the energy eigenfunctions split into two separate manifolds with even and odd symmetry. The height of the barrier determines the energy difference between the lowest even and odd symmetry. This so-called tunnel splitting can be expressed as a tunnel frequency Uf... 

The usefulness of this method is that we can calculate energies from a variety of functions, and we know that the lowest energy obtained is closest to the truth. More systematically, we can express a trial eigenfunction as a function of certain variables and calculate the energy from equation (1.32) as a function of these variables. The energy can be minimized with respect to these variables, and we will then have obtained the best energy value from this type of function. 

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