Matrix form

With constant steps and v = n eq.(5) can be described as linear equation system presented in matrix form ... 

In this method, one notes that real-valued solutions of the time-independent Hamiltonian of a 2 x 2 matrix form can be written in terms of an 0(<1), q), which is twice the mixing angle, such that the electronic component which is initially 1 is cos [0(4>, < )/2], while that which is initially 0 is sin [0(4>,<3r)/2]. For the second matrix form in Eq. (68) (in which, for simplicity f x) = 1), we get... 

Choosing a basis set for the nuclear functions 4>a allows us to write Eq. (52) in a matrix form, similar to Eq. (10) for the single-surface case, now as... 

From the time-dependent Schiddinger equation in the matrix form of Eq. (58), it can be shown [54] that if S(R) is chosen so that... 

By using this expression for G, it is possible to write the nuclear Schrodinger equation (A.8) in matrix form [54,179] as... 

The matrix of a structure with n atoms consists of an array of n / u entries. A molecule with its different atoms and bond types can be represented in matrix form in different ways depending on wbat kind of entries are chosen for the atoms and bonds. Thus, a variety of matrices has been proposed adjacency, distance, incidence, bond, and bond-electron matrices. 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

We have already seen the normal equations in matrix form. In the multivariate case, there are as many slope parameters as there are independent variables and there is one intercept. The simplest multivariate problem is that in which there are only two independent variables and the intercept is zero... 

The rules of matrix-vector multiplication show that the matrix form is the same as the algebraic form, Eq. (5-25)... 

Write the equations of motion in matrix form. Take mi and m2 to be unit masses (say 1.000 kg each). 

Write out the coefficients of the butadienyl system, as they are produced by program MOBAS, in matrix form. Is the matrix symmetric If not, can it be made symmetric by exchanging rows only ... 

One valid form of the input file is the z-matrix form usually associated with GAUSSIAN calculations... 

The set of eigenveetors of any Hermitian matrix form a eomplete set over the spaee they span in the sense that the sum of the projeetion matriees eonstrueted from these eigenveetors gives an exaet representation of the identity matrix. 

Ionization can be improved in many cases by placing the sample in a matrix formed from sinapic acid, nicotinic acid, or other materials. This variant of laser desorption is known as matrix-assisted laser desorption ionization (MALDI). The vaporized acids transfer protons to sample molecules (M) to produce protonated ions [M + H]+. 

Fig. 10. The relationship between a chiral template molecule and the polymeric matrix formed ia the presence of the template molecule.

Aqueous dispersions are used for spinning PTEE fibers. The dispersion is mixed with a matrix-forming medium (116,117) and forced through a spinneret into a coagulating bath. The matrix material is removed by heating and the fibers are sintered and drawn molten to develop their full strength. 

The interface region in a composite is important in determining the ultimate properties of the composite. At the interface a discontinuity occurs in one or more material parameters such as elastic moduli, thermodynamic parameters such as chemical potential, and the coefficient of thermal expansion. The importance of the interface region in composites stems from two main reasons the interface occupies a large area in composites, and in general, the reinforcement and the matrix form a system that is not in thermodynamic equiUbhum. 

Stiffness Matrix Method. The stiffness method can be expressed in matrix form as follows ... 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

Linear Equations in Matrix Form Every set of n nonhomoge-neous linear equations in n unknowns... 

---