First and second derivative matrix elements for the stretching, bending, and torsional energy

1989 ◽  
Vol 10 (1) ◽  
pp. 63-76 ◽  
Author(s):  
Kenneth J. Miller ◽  
Robert J. Hinde ◽  
Janet Anderson
Keyword(s):  
Second Derivative ◽  
Matrix Elements ◽  
Torsional Energy ◽  
Derivative Matrix

ALGEBRAIC EQUATION AND ITERATIVE OPTIMIZATION FOR THE OPTIMIZED EFFECTIVE POTENTIAL IN DENSITY FUNCTIONAL THEORY

2003 ◽  
Vol 02 (04) ◽  
pp. 627-638 ◽  
Author(s):  
QIN WU ◽  
WEITAO YANG
Keyword(s):  
Density Functional Theory ◽  
Algebraic Equation ◽  
Effective Potential ◽  
Density Functional ◽  
Finite Basis ◽  
Basis Set ◽  
Optimization Approach ◽  
Second Derivative ◽  
Functional Theory ◽  
Derivative Matrix

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials.

Traveltime and wavefront curvature calculations in three‐dimensional inhomogeneous layered media with curved interfaces

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1466-1494 ◽  
Author(s):  
H. Gjøystdal ◽  
J. E. Reinhardsen ◽  
B. Ursin
Keyword(s):  
Taylor Series ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Reference Source ◽  
Second Derivative ◽  
Second Derivatives ◽  
Derivative Matrix ◽  
Three Dimensional Models ◽  
Derivatives Of

The seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations. For media with constant velocity and for media with constant velocity gradient, simplified solutions exist. In a general inhomogeneous medium these equations must be solved by numerical approximations. The integration of the ray‐tracing and wavefront curvature equations is then performed by a modified divided difference form of the Adams PECE (Predict‐Evaluate‐Correct‐Evaluate) formulas and local extrapolation. The interfaces between the layers are represented by bicubic splines. The changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas. For three‐dimensional media, two quadratic traveltime approximations have been proposed. Both are based on a Taylor series expansion with reference to a ray from a reference source point to a reference receiver point. The first approximation corresponds to expanding the square of the traveltime in a Taylor series and taking the square root of the result. The second approximation corresponds to expanding the traveltime in a Taylor series. The two traveltime approximations may be expressed in source‐receiver coordinates or in midpoint‐half‐offset coordinates. Simplified expressions are obtained when the reference source and receiver coincide, giving zero‐offset approximations, for which the reference ray is a normal‐incidence ray. A new method is proposed for computing the second derivatives of the normal‐incidence traveltime with respect to the source‐receiver midpoint coordinates. By considering a beam of normal‐incidence rays it is shown that the second‐derivative matrix may be found by computing the wavefront curvature along a reference normal‐incidence ray starting at the reflection point with the wavefront curvature equal to the curvature of the reflecting interface. From this second‐derivative matrix the normal moveout velocity can be computed for any seismic line through the reference source‐receiver midpoint. It is also shown how a reverse wavefront curvature calculation may be used, in a time‐to‐depth migration scheme, to compute the curvature of the reflecting interface from the estimated second derivatives of the normal‐incidence traveltime. Numerical results for different three‐dimensional models indicate that the first traveltime approximation, based on an expansion of the square of the traveltime, is the most accurate for shallow reflectors and for simple models. For deeper reflectors the two approximations give comparable results, and for models with complicated velocity variations the second approximation may be slightly better than the first one, depending on the particular model chosen. A simplified traveltime approximation may be used in a three‐dimensional seismic velocity analysis. Instead of estimating the stacking velocity one must estimate three elements in a [Formula: see text] symmetric matrix. The accuracy and range of validity of the simplified traveltime approximation are investigated for different three‐dimensional models.

Ab initio evaluation of the Born correction, Born couplings, and higher derivative matrix elements with Gaussian‐lobe orbitals

1988 ◽  
Vol 88 (12) ◽  
pp. 7662-7670 ◽  
Author(s):  
Yongfeng Zhang ◽  
Nagamani Sukumar ◽  
Jerry L. Whitten ◽  
Richard N. Porter
Keyword(s):  
Ab Initio ◽  
Matrix Elements ◽  
Higher Derivative ◽  
Derivative Matrix
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