Vectors in Space

In quantum theory, physical systems move in vector spaces that are, unlike those in classical physics, essentially complex. This difference has had considerable impact on the status, interpretation, and mathematics of the theory. These aspects will be discussed in this chapter within the general context of simple molecular systems, while concentrating at the same time on instances in which the electronic states of the molecule are exactly or neatly degenerate. It is hoped... 

L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces Second Edition (1995)... 

The product c a) is another ket, and c a) = a)c. It is postulated that a) and c a), (c 0) represent the same physical state, and only the direction in vector space is of significance. This is more like the property of a ray than a vector. 

We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eb the squared length l2 of a n-vector t> is the quadratic form given by... 

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

The Patterson map, commonly designated P(uvw), is a Fourier synthesis that uses the indices, h,k,l, and the square of the structure factor amplitude, F(hkl), of each diffracted beam. It is usual to describe the Patterson map in vector space defined by u, v, and w, rather than x,y,z as used in electron-density maps. 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

The idea of a vector triangle provided the eventual solution to the phase problem. Three associated vectors with appropriately linked properties should be able to form a triangle in vector space (Figure 6.6). Given the fact that each F(hkl) structure factor, corresponding to a given set of h/cHattice planes, is also a vector in the complex plane, then the idea was... 

Many of the vectors employed in chemical informatic applications must be viewed as geometric rather than as algebraic objects since they do not satisfy the vector space axioms. Since the component values of molecular vectors are, except in rare cases, positive their associated vectors will lie in the positive hyper-quadrant. Subtracting two such vectors, an operation that is allowed in vector spaces, may yield a vector that does not lie within the positive hyper-quadrant, and thus, does not correspond to any molecule that can be represented by that form of representation. For example, vectors whose components are BCUT descriptors [74, 75] do not satisfy the vector-space axioms, since adding two such vectors may produce a vector that lies outside of BCUT chemical space. This does not, however, pose a practical problem since such vectors can be thought of as entities that describe points in a geometric space not vectors in a vector space. 

Generally, if g is a function defined on P, let g(X) be the (random) variable associating the value g(x) with any value x of X in vector space Then the integral mean value of g, with density /x, equals by definition... 

The Danbechies 4 wavelet fnnction reqnires that the input image be a square with a dimension of 2 X where j is an integer. We determined that an ANN with 4096 neurons in the input layer is the largest ANN that can be trained in a reasonable amount of time on a Sun Blade 100 or a Pentium 4 computer, which were the computers used in this part of the project. This means that wavelet coefhcients in vector spaces Vq, Vj, V2, V3, V4 and V5 were used, producing an input matrix with a dimension of 2 x 2 . This size inpnt matrix can only be generated from an image scaled to 256 X 256 pixels (2 x 2 ) prior to Danbechies 4 encoding. 

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

In Liouville space, both the density matrix and the operator are vectors. The dot product of these Liouville space... 

The coefficients p. are chosen so that, on a quadratic surface, the interpolated gradient becomes orthogonal to all Aq. This condition is equivalent to minimizing the energy in the space spaimed by the displacement vectors. In the quadratic case, a further simplification can be made as it can be shown that all p. with the... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Throughout, the space coordinates and other vectorial quantities are written either in vector fomi x, or with Latin indices k— 1,2,3) the time it) coordinate is Ap = ct. A four vector will have Greek lettered indices, such as Xv (v = 0,1,2,3) or the partial derivatives 0v- m is the electronic mass, and e the charge. 

Figure 6. Two-dimensional (top) and 3D (bottom) representations of a peaked (a) and sloped (b) conical intersection topology. There are two directions that lift the degeneracy the GD and the DC. The top figures have energy plotted against the DC while the bottom figures represent the energy plotted in the space of hoth the GD and DC vectors. At a peaked intersection, as shown at the bottom of (a), the probability of recrossing the conical intersection should be small whereas in the case of a sloped intersection [bottom of ( )l, this possibility should be high. [Reproduced from [84] courtesy of Elsevier Publishers.]...

In general, a vector in an n-space can be represented by an n-tuple of numbers for example, a vector in 3-space can be represented as a number triplet. 

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