Cartesian angular momentum

The elements V2,-m describe the spatial interaction in a reference axis system of choice and are connected to Qz.m by a rotation as shown in Eq. (4). The spin tensor elements 72, are given in terms of the Cartesian angular momentum operators by t... 

Table 4.11 Angular momentum operators Lx, Ly, acting on cartesian angular momentum harmonic eigenfunctions in first column produce eigenfunctions of the same type with 1 = 2...

The generalization by mathematical induction is described next, without including the proof. In fact, the application of the three operators p , Py, on the 2 + 1 linearly independent cartesian angular momentum eigenfunctions multiplied by the radial Bessel function Zt ikr) lead to the following results for even and odd, respectively. 

Similarly, of the 3(4n + 3) combinations for = 2n + l, there are 2(2n + 2) +1 with = 2n + 2 and 2(2n) +1 with = 2n linearly independent combinations, which can be identified with cartesian angular momentum eigenfunctions of even degrees 2n + 2 and 2n, respectively, multiplied by their respective 2a,+2(A r) and Z2n(kr) radial functions. The combinations for the latter are also identified through their additional common factor + y + 7 =... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

The third quantum number m is called the magnetic quantum number for it is only in an applied magnetic field that it is possible to define a direction within the atom with respect to which the orbital can be directed. In general, the magnetic quantum number can take up 2/ + 1 values (i.e. 0, 1,. .., /) thus an s electron (which is spherically symmetrical and has zero orbital angular momentum) can have only one orientation, but a p electron can have three (frequently chosen to be the jc, y, and z directions in Cartesian coordinates). Likewise there are five possibilities for d orbitals and seven for f orbitals. 

The form of Lz in Cartesian coordinates is Eq. (7-la), and it is clear that orbital angular momentum is related to angular displacement in the same way as the linear operators are related. 

Here Uj and Uj are Cartesian unit vectors, a) and j3) are localized orbitals that are doubly occupied in the HF ground state, jm) and n) are virtual orbitals. Rq is the position vector of the local gauge origin assigned to orbital a) and = (r — R ) x p is the angular momentum relative to Re- Superscript 1 denotes terms to first order in the fluctuation potential, and = [A — is the principal propagator at the zero energy... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

N is a normalization factor which ensures that = 1 (but note that the are not orthogonal, i. e., 0 lor p v). a represents the orbital exponent which determines how compact (large a) or diffuse (small a) the resulting function is. L = 1 + m + n is used to classify the GTO as s-functions (L = 0), p-functions (L = 1), d-functions (L = 2), etc. Note, however, that for L > 1 the number of cartesian GTO functions exceeds the number of (27+1) physical functions of angular momentum l. For example, among the six cartesian functions with L = 2, one is spherically symmetric and is therefore not a d-type, but an s-function. Similarly the ten cartesian L = 3 functions include an unwanted set of three p-type functions. 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

Polarization functions may be optionally chosen to be of pure or Cartesian form (by another keyword). In the former case, one includes the expected number of angular-momentum components (i.e., five d orbitals, seven f orbitals, etc.), whereas in the latter case some additional component(s) of lower angular momentum are included (e.g., a Cartesian d set includes five d orbitals plus one s orbital, a Cartesian f set includes seven f orbitals plus three p orbitals, and so forth). 

Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations... 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

As one increases the indexing, the disparity between the number of Cartesian functions and the number of canonical functions increases. Thus, with f-type GTOs (indices summing to 3) there are 10 Cartesian functions and 7 canonical functions, with g-type 15 and 10, etc. GTOs can be taken arbitrarily high in angular momentum. 

Equation (13) shows that the complete temperature and field dependence of the strains can be calculated from static correlation functions (J Jj )7-,h (y, y — 1.2,3 label the cartesian components of the angular momentum J) where O7- h denote thermal expectation values (Callen and Callen 1965). As already mentioned above, a mean field theory may be used to evaluate (13) and calculate the magnetostriction. 

The angular momentum operator is a first-rank tensor whose Cartesian and spherical components transform into each other as... 

The matrix elements of the angular momentum operators in Cartesian directions a = x,y, z form (complex) matrices La. Now we can proceed with an evaluation of the /1-tensor components in three steps. 

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