Unitary coordinate system

Nineteenth-century crystallography may be considered to be the mathematical branch of mineralogy. It is based on two empirical laws, the law of constancy of angle and the law of rational indices. These laws will be presented in the following pages after a discussion of some mathematical principles fundamental to crystallography, non-unitary coordinate systems and reciprocal coordinates. 

By choosing two non-collinear translations a and b in Fig. 1.8(a), we describe the lattice by the translation vectors r = wa H- t b,M and v being integers. We call this coordinate system the lattice base. The parallelogram (a, b) is the cell (unit cell). Analogously, the base a, b, c of a three-dimensional lattice is defined by three non-coplanar translations. The cell is hence a parallelepiped. The coordinates x, y, z of a point inside this cell are referred to this non-unitary coordinate system. The set of all the points equivalent by translation to the point Xpyp Zj is given by... 

If (R, t) is a symmetry operation, the norms of r and r in Figs 2.1 and 2.2 are equal. By choosing a unitary coordinate system, we obtain from this condition that IIr II2 = r R Rr= rp, thus R R = E = unit matrix (R is the transpose of the matrix R). This leads to the result that R is an orthogonal matrix ... 

We have shown that R is represented by an orthogonal matrix if we choose a unitary coordinate system. The matrices (2.7), (2.8) and (2.9) are examples of orthogonal matrices. Alternatively, we can choose the axes of our coordinate system to be a lattice base a, b, c, i.e. three primitive non-coplanar translations. The coordinates of the lattice points u, u, w are then integers and all the terms in the corresponding representation of R are thus also integers. Let us indicate the orthogonal representation by the matrix U, and the representation with integers by the matrix N. The matrices U and N are related by a similarity transformation because they represent the same operation. Hence, there must exist a matrix X such that N = X" UX. The matrix X transforms the coordinate system of the lattice to a unitary system. Moreover, U is similar to one of the matrices (2.7), (2.8) or (2.9). We know that similar matrices have the same trace. It thus follows that ... 

In order to avoid the mathematical difficulties associated with the metric (Section 1.2), in crystal physics we use only unitary coordinate systems three mutually perpendicular axes 61,62, 63 of length e, = 1. In this coordinate system, a vector A is represented by a 3 x 1 matrix (a column vector) the transposed representation is a line vector. 

A change in coordinate system is represented by an orthogonal transformation matrix U (unitary coordinate system),... 

The axes of the unitary coordinate system are chosen in the following manner parallel to a, 02 parallel to b, 63 parallel to c. The reciprocal vectors are given in Exercise 5.4.2. The reciprocal vector a + 4c = + 4c C3 is coincident with the normal to the wave v. The... 

Certain special features to be imposed on a model may be expressed by more complicated constraint equations. We note as an example the assumption of a rigid molecule with prescribed dimensions whose position and orientation are to be refined. The position may be described by the coordinates of the centre of mass and the orientation by three Euler angles with respect to a unitary coordinate system. The atomic coordinates and thus the structure factor. Equation [1], are expressed as functions of these six parameters. The latter may then be adjusted to optimize the deviance. A similar procedure can be used to constrain the atomic displacement parameters of a molecule to rigid-body movements described by a translation tensor, a libration tensor and a transla-tion/libration-correlation tensor (TLS model). This model neglects intramolecular vibrations. 

The physical meaning of the sections on transformation matrices and unitary matrices is that we can try to rotate our coordinate system so that each component... 

If the coordinate system has been transformed to the normal coordinate system by a unitary transformation U, the Hessian is diagonal and... 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

In the coordinate system (x ) where the Hessian is diagonal (i.e. performing a unitary transformation, see Chapter 13), the NR step may be written as... 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

The special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

The coordinate systems chosen in crystallography are generally defined by three nonorthogonal base vectors a,b,c of different lengths (a,6,c). These non-unitary systems introduce some complexity into the expressions used in analytical geometry,... 

The rotation matrix is a unitary (orthogonal) matrix U, since the determinant is equal to 1 (cos or-t sin a= 1). The significance of a unitary matrix is that it describes a rotation of the coordinate system without changing the length of the coordinate axes. A unitary matrix with a determinant of -1 describes a rotation of the coordinate system, followed by inverting the directions of the coordinate axis, i.e. an improper rotation in the language of point group symmetry. 

Changing the coordinate system thus changes a matrix by pre- and post-multiplication of a unitary matrix and its inverse, a procedure called a similarity transformation. Since the U matrix describes a rotation of the coordinate system in an arbitrary direction, one person s U may be another person s U . There is thus no significance whether the transformation is written as U AU or UAU and for an orthogonal transformation matrix (U = U ), the transformation may also be written as IPAU or UAU . 

We will need the unitary transformations exp (lA) and exp (iS). They are very convenient, since when starting from some set of the orthonormal functions (spinorbitals or Slater determinants) and applying this transformation, we always retain the orthonormality of new spinorbitals (due to A) and of the linear combination of determinants (due to S). This is an analogy to the rotation of the Cartesian coordinate system. It follows from the above equations that exp (/A) modifies spinorbitals (i.e., operates in the one-electron space), and exp (i S) rotates the determinants in the space of many-electron functions. 

The values of the coefficients will depend on the geometry of the system. However, there are certain relations involving the which are independent of the geometry. Equation (IIIB-5) is equi valent to a rotation of axes from the coordinate system (i = 1. . . iV) to the coordinate system j. (if = 1. . . Af) therefore, the are components of a unitary matrix (Eyring, Walter,... 

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