Diagonalization-least squares procedure

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

The block of 10 by 10 cells to be upscaled is isolated along with a skin. The flow equation is solved using the boundary conditions shown in the Figure. These boundary conditions force flow in four different directions. Then, a simple least-square procedure is used to determine the single conductivity tensor that is able to reproduce best the flows crossing the block for all four boundary conditions. The result is a full hydraulic conductivity tensor with arbitrary principal directions, not necessarily parallel to the Cartesian axes. In Figure 4, the diagonal components of the... 

The simulation procedure presented here, among others, is eminently exploitable to estimation of spin-Hamiltonian parameters from a powder spectrum by the least-squares fitting (LSF) procedure in conjunction with numerical diagonalization of the spin-Hamiltonian matrix similar to that proposed by Misra (1976) in context... 

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