Visualizing Atomic Orbitals Jing B. Wang, Paul C. Abbott, and Jim F. Williams Department of Physics University of Western Australia Nedlands WA 6907, Australia
[email protected],
[email protected]
The Mathematica software package provides a range of tools for working with atomic systems. The symbolic tools include orthogonal polynomials and Clebsch-Gordan coefficients whilst the graphical capabilities cover polar plots; spherical plots; density plots; contour plots in two and three dimensions; and animation. These tools are applied to the manipulation and visualization of atomic orbitals.
INTRODUCTION The probability distribution of the electron charge cloud in an atom, namely the atomic orbital, often reveals crucial information in atomic physics. Visual representation of such orbitals is invaluable to researchers working in this field and also helps teaching atomic structures to physics or chemistry students. The atomic orbital concepts were discussed in detail by Berry and the cited papers therein, including several visualization devices. However, the newly developed symbolic package Mathematica provides a wider range of graphical tools (including polar plots, spherical plots, density plots, contour plots in two and three dimensions, and animation), which can be applied to help visualise the radial and angular behaviour of multi-dimensional systems. Mathematica also has a wide range of built-in special functions appropriate for the analysis of quantum systems. Orthogonal polynomials — Legendre, Laguerre, Chebyshev, Hermite and Jacobi (corresponding to the Mathematx) — arise when solving a wide class of eigenvalue and ica functions Pn x, Lan x, Un x, Tn x, Hn x, and Pa,b n boundary value problems. The spherical harmonics, Ylm Θ, Φ, are eigenfunctions of the angular part of the Laplacian operator (which is the quantum mechanical angular momentum operator). Clebsch-Gordan coefficients arise in the study of angular momenta in quantum mechanics, and in other applications of the rotation group. These functions, taken together with the orthogonal polynomials and graphical tools, provide a powerful environment for the study of atomic systems. It is worth noting that these methods can be readily utilized in the studies of other multi-dimensional systems. As this article is not intended to be a tutorial on Mathematica, commands and syntax will not usually be explained. However, it is hoped that by showing a well-chosen set of examples, even readers unfamiliar with Mathematica will be able to follow the examples. The graphical output is placed immediately after its corresponding input command as it appears in Mathematica notebooks. Arbitrary units are used in these graphs.
1. NOTATION AND PREREQUISITES The quantity pure basis state Ψn,l,m r, Ψn,l,m r r r r Ψ r r coherently mixed state Ψ r c Ψ n,l n,l m m n,l,m r, where Ψn,l,m r n l m Rn,l rYlm Θ, Φ, r r, Θ, Φ, r r2 r , sinΘ Θ Φ, represents the probability of finding an electron in the volume element r. The spherical harmonics, Ylm Θ, Φ, give the angular distribution of the atomic orbitals and Rn,l r is the corresponding radial distribution. In the following, the general spherical harmonic will be denoted using the Mathematica shorthand Ylm Θ, Φ. For example,
2
Wang, Abbott, and Williams Y21 Θ,
Φ
15 1
cosΘ sinΘ
Φ 2Π 2
Complex conjugation, required to yield real-valued charge densities, is easily achieved using a replacement rule: exp_ : exp . Complexa_, b_ Complexa, b
For example,
Y21 Θ, Φ
15 1
cosΘ sinΘ
Φ 2Π 2
Several Mathematica graphics packages are used in this work. Entering Graphics`
causes these packages to be automatically loaded. We also set options for nice formatting of graphics labels, $FormatType TraditionalForm;
$TextStyle TraditionalForm, FontSize 10, FontFamily "Times";
and for “pretty” plots: SetOptionsDensityPlot, Mesh False, PlotPoints 200;
SetOptionsParametricPlot3D, PlotPoints 40, PlotRange All, Axes True, AxesLabel x, y, z, Ticks None, Boxed True, BoxStyle Dashing0.001, 0.02;
SetOptionsContourPlot3D, PlotPoints 4, 6;
We make use of the Notation package to define custom input and output notations for the 3- j, 6- j and ClebschGordan coefficients. Utilities`Notation`AutoLoadNotationPalette False;
Utilities`Notation`
SetOptionsNotation, WorkingForm TraditionalForm;
Visualizing Atomic Orbitals
3
Symbolize j
OffClebschGordan::"tri"
OffClebschGordan::"phy"
j1_ j2_ j3_ Notation
m1_ m2_ m3_
Notation
ThreeJSymbolj1_, m1_, j2_, m2_, j3_, m3_
3 j
j1_ j2_ j3_ SixJSymbolj1_, j2_, j3_, j4_, j5_, j6_ j4_ j5_ j6_ 6 j
Notationj1_, j2_,m1_,m2_ j_,m_ ClebschGordanj1_, m1_, j2_, m2_, j_, m_
Here is a palette for entering these special notations.
3 j
6 j
, , , ,
2. ANGULAR DISTRIBUTION OF ATOMIC ORBITALS 1. Pure states Ψnlm r For pure basis states, Ψn,l,m r, the angular charge density distribution: 2 m ang l,m Θ Yl Θ, Φ ,
is axially symmetric with respect to the azimuthal angle Φ. In this case, PolarPlot can provide a complete view of the atomic orbitals. After defining the angular distribution function:
l_,m_ Θ_ : Ylm Θ, Φ Ylm Θ, Φ
individual charge densities can be immediately examined, e.g.,
4
Wang, Abbott, and Williams 3,2 Θ 105 cos2 Θ sin4 Θ
32 Π
Here are plots of the atomic orbitals for the states l 1, m 0, 1: DisplayTogetherPolarPlot 1,0 Θ, Θ, 0, 2 Π, PolarPlot 1,1 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.01, 0.01;
0.1 0.05
0.2
0.1
0.1
0.2
0.05 0.1
and l 2, m 0, 1, 2: DisplayTogetherPolarPlot 2,0 Θ, Θ, 0, 2 Π, PolarPlot 2,1 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.015, 0.015, PolarPlot 2,2 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.003, 0.01;
0.15 0.1 0.05 0.4
0.2
0.2
0.05
0.4
0.1 0.15
0.3 0.3 , DisplayTogether PolarPlot 2,0 Θ, Θ, 0, 2 Π, PlotRange
0.16 0.16
PolarPlot 2,1 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.015, 0.015, PolarPlot 2,2 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.003, 0.01;
0.15 0.1 0.05 0.2
0.1
0.1 0.05 0.1 0.15
0.2
Visualizing Atomic Orbitals
5
0.3 0.3 DisplayTogether PolarPlot 3,0 Θ, Θ, 0, 2 Π, PlotRange ,
0.18 0 .18
PolarPlot 3,1 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.015, 0.015, PolarPlot 3,2 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.008, 0.01, PolarPlot 3,3 Θ, Θ, 0, 2 Π, PlotStyle Dashing0.003, 0.01;
0.2
0.1 0.025 0.05 0.075 0.1 0.125 0.15 0.175
0.1
0.2
2. Mixed states m cm Ψnlm r For coherently mixed states Ψnl r cm Ψnlm r r Rnl r cm Ylm Θ, Φ, m
m
the angular probability distribution is 2 m Φ cm Yl Θ, Φ . m In this case, the charge cloud is no longer axially symmetric and its distribution is a function of both Θ and Φ. Threedimensional plots are thus required to depict the structure of the coherently mixed atomic orbitals. ang l Θ,
In general, the expression l
Ωl Θ, Φ am bm Ylm Θ, Φ, m l,2
where am (bm ) are the amplitude ratio (relative phase) of each state, can be evaluated in Mathematica using:
l
Ωl_ c_Θ_, Φ_ : Ωl cΘ, Φ ci1,1 ci1,2 Yll2 i Θ, Φ i0
The argument to c needs to be a list of am , bm pairs. The general angular charge distribution then reads: l_ c_Θ_, Φ_ : FactorComplexExpandExpandΩl cΘ, Φ Ωl cΘ, Φ
where the core of this expression is the product of Ωl cΘ, Φ with its complex conjugate, Ωl c Θ, Φ . The other operations, Expand, ComplexExpand, and Factor, are used to simplify the expression to a human readable form. As an example, for the general P-state wavefunction can be obtained from the mixed state consisting of l 1, m 1: 1 0 Θ, Φ ΨP a_, b_ 1
a b 3 a2 2 cosb 2 Φ a 1 sin2 Θ
8Π
and then plot the orbitals for the states with parameters a 0.1, b 0:
6
Wang, Abbott, and Williams SphericalPlot3DEvaluateΨP 0.1, 0, Θ, 0, Π, Φ, 0, 2 Π;
z
y
x
and a 0.2, b Π 3: Π SphericalPlot3DEvaluateΨP 0.2, , Θ, 0, Π, Φ, 0, 1.35 Π; 3
z
y
x
Note that the full range of Φ was not shown to highlight the structure of the orbital. For a more complicated atomic orbitals, such as a coherently excited F-state atom, the charge density reads
Visualizing Atomic Orbitals
7
a3 b3 a1 0 Θ, Φ; ΨF Θ, Φ ΨF Θ_, Φ_ : 3 a1 b1
a3 b3 1
7 sin2 Θ 75 a2 1 cos4 Θ 75 a21 cos4 Θ 150 cos2 Φ b 1 a 1 a1 cos4 Θ 64 Π 30 a2 1 cos2 Θ 30 a21 cos2 Θ 10 15 cos2 Φ b 3 b 1 sin2 Θ a 3 a 1 cos2 Θ 10 15 cos4 Φ b 3 sin2 Θ a 3 a1 cos2 Θ 60 cos2 Φ b 1 a 1 a1 cos2 Θ 10 15 cos4 Φ b 1 b3 sin2 Θ a 1 a3 cos2 Θ 10 15 cos2 Φ b3 sin2 Θ a1 a3 cos2 Θ 5 sin4 Θ a2 3 3 a2 1 3 a21 5 sin4 Θ a23 2 15 cos2 Φ b 3 b 1 sin2 Θ a 3 a 1 2 2 15 cos4 Φ b 3 sin Θ a 3 a1 6 cos2 Φ b 1 a 1 a1 10 cos6 Φ b 3 b3 sin4 Θ a 3 a3 2 15 cos4 Φ b 1 b3 sin2 Θ a 1 a3 2 15 cos2 Φ b3 sin2 Θ a1 a3
Choosing the parameters arbitrarily as a : a1 1.0; a : a1 0.0; a : a3 0.8; a : a3 0.05;
Π Π b : b1 Π; b : b3 ; b : b3 ; 5 4
one can produce a three-dimensional graph: SphericalPlot3DEvaluateΨF Θ, Φ, Θ, 0, Π, Φ, 0, 2 Π, PlotPoints 50, PlotRange All;
y
z
x
We can use the Shadow command to display the projections along the x, y, and z directions:
8
Wang, Abbott, and Williams Shadow%;
3. Time evolution of atomic orbitals Due to the interaction between the intrinsic magnetic dipole moment of the electron (spin) and the internal magnetic field of the atom (orbital angular momentum), the atomic charge density evolves with time. Animation provides views of such dynamic effects. Taking the spin-orbit (L-S) coupling into account, we obtained the timedependent charge density function for a mixed state: Ψnl r cm Ψnlm r, m
l l K ang nl Θ, Φ, t 1l m 2K 1 Ylm Θ, ΦYlm Θ, Φ Tl†KQ GK ls; t, m m Q KQmm
where l l K Tl†KQ 1l m 2K 1 m Q m mm
cm cm
implemented in Mathematica as
l
K_,Q_ l_ : K,Q l 1lmQ ml
l K l cmQ c 2 K 1 m
m Q m Q 3 j
are the so-called state multipoles and the corresponding time evolution factor is l j s 2 1 t ! cos j j
. GK ls; t
2 j 12 j 1 j l K 2s 1 j j
implemented in Mathematica as
Visualizing Atomic Orbitals
9 ls
ls
1 l GK_ l_, s_, t_ : G K l, s, t 2 j 1 2 j 1 j 2s1 j ls j ls
j s cos j j t l K 6 j 2
As an example, consider a general mixed P-state of a hydrogen atom: 1 l 1; s ; c : c1 a b ; c : c1 d e ; c : c0 f g ; 2
The time-dependent charge density function becomes
CollectComplexExpand 2l
K
l
lmQ K,Q l GK l, s, t 1 K0
QK ml
l K l YlmQ Θ, Φ Ym Θ, Φ , 2 K 1 l
m Q m Q 3 j
cosΘ, sinΘ, cos2 Φ, sin2 Φ, Simplify 2 a2 2 b2 2 d 2 2 e2 5 f 2 5 g2 2 a2 b2 d 2 e2 2 f 2 2 g2 cost cos2 Θ
12 Π 2 cost 1 sinΘ a f d f b e g cosΦ b f e f a d g sinΦ cosΘ
2 2 Π 7 a2 7 b2 7 d2 7 e2 4 f 2 4 g2 2 a2 b2 d2 e2 2 f 2 2 g2 cost sin2 Θ
24 Π a d b e 2 cost 1 cos2 Φ b d a e 2 cost 1 sin2 Φ
4Π 4Π
In a particular case, say C0 0, C1 0.9, C 1 0.12 0.1 , the charge density function is cdft_ Chop% . a 0.9, b 0, d 0.12, e 0.1, f 0, g 0;
This can be plotted using the commands, CDFPlott_ : SphericalPlot3D Evaluatecdft, Θ, 0, Π, Φ, 0, 2 Π,
0.14 0.14 PlotPoints 30, PlotRange 0.14 0.14 , Boxed False, Axes False;
0.12 0.12
tΠ TableCDFPlot , t, 0, 5, 1;
5
This command displays a sequence of graphics objects in succession and thus simulates the time evolution of the orbitals. Alternatively, here is an array of animation frames:
10
Wang, Abbott, and Williams ShowGraphicsArrayPartition%, 3, DisplayFunction $DisplayFunction;
We clear the values assigned in this section.
Clearl, s, c;
3. RADIAL AND ANGULAR DISTRIBUTIONS 1. Radial function For the hydrogen atom the radial wavefunction in atomic units is 2Z 3 n l 1 Ρ2 l 2 l1
Ρ Ln l 1 Ρ, Rn,l r n 2n n l where 2Zr Ρ ; n
and LΑn x denote the associated Laguerre Polynomials as defined by Abramowitz and Stegun. Although the definition of the associated Laguerre Polynomials differs from that used in many quantum mechanics texts, it has several advantages: it is the standard used by most mathematicians and several books of tables; the number of nodes of the radial wavefunction equals the order of the associated Laguerre Polynomial, i.e., n l 1; and it is the definition used in Mathematica. The radial wavefunction is easily defined:
3 1 2 Z n l """""""""" ; n_,l_,Z_: 1 !""""""""""""""""""""""""""""""""
n 2 n l n
Ρ
l1 Rn_,l_,Z_: 1 r_ n,l,Z 2 Ρl L2nl1 Ρ;
The syntax, Z_: 1, indicates that if the argument Z is not supplied it will default to the value 1. The radial distribution function,
Visualizing Atomic Orbitals
11
n,l r r2 Rn,l r2 ,
then becomes n_,l_ r_ : r 2 Rn,l r2
Here is a plot of three states with n 2, 3, 4, l 1: DisplayTogetherPlotEvaluate1,0 r, r, 0, 40, PlotRange All, PlotEvaluate2,0 r, r, 0, 40, PlotStyle Dashing0.008, 0.005, 0.01, PlotEvaluate3,0 r, r, 0, 40, PlotStyle Dashing0.005, 0.01, PlotEvaluate4,0 r, r, 0, 40, PlotStyle Dashing0.002, 0.01, 0.002;
0.5 0.4 0.3 0.2 0.1
10
20
30
40
2. Density plots for pure basis states We have so far plotted separately the angular and the radial distribution functions of various atomic orbitals. However, these pictures are incomplete on their own and one needs to imaginatively assemble the angular and the radial plots in order to obtain a complete picture of the charge distribution in three-dimensional space. For the pure basis state the probability distribution nlm r is, as discussed earlier, axially symmetric, i.e., nlm r, Θ r Rnl rYlm Θ, Φ2 . Mathematica's DensityPlot is then adequate for showing the detailed distribution of the two-variable function, where the lighter region indicates higher charge density: n_,l_,m_ r_, Θ_ : n,l r l,m Θ
Defining a set of rules to transform from spherical polar to Cartesian coordinates:
PolarToCartesian r
z y #$$$$$$$$$$$$$$$$ $$$$$$$$$ x2 y2 z2 , Θ cos1 , Φ tan1 ; r x
the distribution in Cartesian coordinates is easily obtained: n_,l_,m_ x_, y_, z_ n,l,m r, Θ . PolarToCartesian;
Here are plots of the charge density in the x-z plane for n 2, 3, 4, l 1, m 0:
12
Wang, Abbott, and Williams DensityPlotEvaluate2,1,0 x, 0, z, x, 10, 10, z, 10, 10;
10
5
0
5
10 10
5
0
5
10
10
20
DensityPlotEvaluate3,1,0 x, 0, z, x, 20, 20, z, 20, 20;
20
10
0
10
20 20
10
0
Visualizing Atomic Orbitals
13
DensityPlotEvaluate4,1,0 x, 0, z, x, 40, 40, z, 40, 40;
40
20
0
20
40 40
20
0
20
40
3. Three dimensional contour plots for mixed states Finally, for a general coherent mixture of several magnetic states Ψnl r cm Ψnlm r r Rnl r cm Ylm Θ, Φ, m
m
the total charge density distribution is a function of three variables (r,Θ,Φ): n_,l_ c_r_, Θ_, Φ_ : n,l r l cΘ, Φ 2 Arbitrarily choosing C1 1, C0 0, C 1
10
, three dimensional contour surfaces of the total charge distribution in Cartesian coordinates,
1 0 r, Θ, Φ . PolarToCartesian TrigExpand Simplify2,1 2 0
10 2 2 2
x y z 4 x2 9 y2 x2 y2 z2
400 Π
can be plotted using ContourPlot3D. Here are the surfaces for the contour levels 0.0003, 0.0015, and 0.003.
14
Wang, Abbott, and Williams ContourPlot3D%, x, 10, 0, y, 10, 10, z, 10, 10, Contours 0.0003, 0.0015, 0.003;
4. MOLECULAR ORBITALS As an example of multi-dimensional systems other than atomic orbitals, we consider the hydrogen molecule-ion, which consists of two nuclei and one electron. Its wavefunction can be represented by a linear combination of two atomic hydrogen wavefunctions: 1
Φ1 x r0 , y, z Φ2 x r0 , y, z bondingorbitals,
2 1s Ψ ci Φi 1
Φ1 x r0 , y, z Φ2 x r0 , y, z antibondingorbitals, i 2 1 s where 2r0 is the internuclear distance and S is a function of r0 . In Mathematica syntax, we have for the atomic wavefunction,
Φn_,l_,m_ x_, y_, z_ r Rn,l r Ylm Θ, Φ . PolarToCartesian;
and for the molecular charge density of the bonding, n1_ l1_ m1_ ΨB x_, y_, z_ Φn1,l1,m1 x 2, y, z Φn2,l2,m2 x 2, y, z2 ;
n2_ l2_ m2_
and antibonding orbitals, n1_ l1_ m1_ x_, y_, z_ Φn1,l1,m1 x 2, y, z Φn2,l2,m2 x 2, y, z2 ; ΨA
n2_ l2_ m2_
If the hydrogen molecular-ion is in the state n1 1, l1 0, m1 0, n2 1, l2 0, m2 0, both bonding and antibonding orbitals have cylindrical symmetry along the internuclear x-z plane. The DensityPlot command is used to depict the projection of the charge distribution in this plane.
Visualizing Atomic Orbitals
15
1 0 0 DensityPlotEvaluateΨB x, 0, z, x, 5, 5, z, 5, 5;
1 0 0
4
2
0
2
4
4
2
0
2
4
1 0 0 DensityPlotEvaluateΨA x, 0, z, x, 5, 5, z, 5, 5;
1 0 0
4
2
0
2
4
4
2
0
2
4
To see these orbitals in three dimensions, we also plot their contour surface at an arbitrary level of 0.005:
16
Wang, Abbott, and Williams 1 0 0 ContourPlot3DEvaluateΨB x, y, z, x, 5, 5, y, 5, 5, z, 5, 5, Contours 0.005;
1 0 0
%% %% 1 0 0 ContourPlot3DEvaluate %Ψ A x, y, z%, x, 5, 5, y, 5, 5, z, 5, 5, Contours 0.005; %
1 0 0 %
5. EXAMPLES OF SYMBOLIC ANALYSIS 1. Clebsch-Gordan coefficients Clebsch-Gordan, 3- j, 6- j and 9- j coefficients (CGC) arise when coupling together angular momentum states or, equivalently, when linearizing products of or computing integrals over spherical harmonics. It is important to note that Mathematica can compute both numerical and symbolic CGC.
AdvantagesAtomic of symbolic Visualizing Orbitals computation of CGC include evaluating whole classes of related coefficients in one operation, and easy conversion to fortran or C (using FortranForm or CForm), if rapid computation is required. The notation l1 l2 m1 m2 " lm C lm1 lm2 lm 1ml1 l2 2 l 1 lm1 lm2 l m , 1 2 1 2 is used in the following examples of the utility of symbolic CGC. The CGC required for spin-orbit coupling can be evaluated immediately, e.g., 1 1 1 %%% 1 & j1 , ,m , %%%% j1 ,m' Simplify 2 2 2 %% 2 2 m 2 j1 1
4 j1 2
2. Products and integrals of spherical harmonics Products of spherical harmonics arise when computing the electron density of atomic states. Using the CGC, it is straightforward to linearize these products via l1 l2
Ylm1 1
Θ,
Φ Ylm2 2
l3
2l1 12l2 12l3 1 l1 l2 l3 l1 l2 l3
Θ, Φ Y m3 Θ, Φ . 0 0 0 m1 m 2 m 3 l 3 4Π l3 l1 l2 m l3
This result can be used to evaluate the general integral of a triple product of spherical harmonics, Π
2Π
#
0
# Ylm1 1 Θ, Φ Ylm2 2 Θ, Φ Ylm3 3 Θ, Φ sinΘ Θ Φ 0
2l1 12l2 12l3 1 l1 l2 l3 l1 l2 l3
. 0 0 0 m1 m2 m3 4Π In Mathematica this formula can be entered as follows: TripleProductl1_, m1_, l2_, m2_, l3_, m3_ : 2 l1 1 2 l2 )))))))))))))))) 1 2 l3)))))))) 1 l1 l2 l3 l1 l2 l3 ()))))))))))))))))))))))))))))))) )))) 4 Π
0 0 0 3 j m1 m2 m3 3 j
Notation&
j1_ j2_ j3_ ' TripleProductj1_, m1_, j2_, m2_, j3_, m3_ m1_ m2_ m3_
This expression is used in the following example of the linear Stark effect.
3. Stark effect The linear Stark effect requires the computation of the matrix elements n l mr cosΘn l 1m . Noting that cosΘ
Π Y 0 Θ, Φ, is equivalent to 2 $%%%%% 3
1
Π 2 ()))))) Y10 Θ, Φ cosΘ 3 True
17
18
and that
Wang, Abbott, and Williams
Ylm Θ, Φ Ylm Θ, Φ 1m Yl m Θ, Φ, the required non-vanishing angular integrals, i.e., l m " cosΘ " l 1m , are easily evaluated. Π l 1 l1 Simplify2 ()))))) 1m & ', l, m m 0 m 3 l m1 lm1
4 l2 8 l 3
Π l 1 l 1 Simplify2 ()))))) 1m & ', l, m m 0 m 3 l m lm
4 l2 1
6. CONCLUDING REMARKS This article has demonstrated the capabilities of Mathematica for analyzing and visualizing atomic systems. The interplay between symbolic and graphical methods yields a tool that is more powerful than the sum of its parts and one that can be superior to special purpose visualization tools because of its flexibility. Even for readers that have never used a computer algebra system, an examination of some of the code fragments presented above — say those implementing the radial wavefunctions for hydrogenic atoms or the triple integral of spherical harmonics — illustrates that implementing some quite complicated formulae in Mathematica is straightforward and transparent and, hence, less likely to contain or conceal an error. One of the goals of object oriented programming is to implement high-level structures. All of the computations in this article could, of course, be done in fortran or C (or C++). However, in general, the physicist or chemist does not want to write packages for computing orthogonal polynomials, spherical harmonics, Clebsch-Gordan coefficients or basic graphical tools. Of course, there exists libraries of public domain and commercial software which, when linked together, can provide equivalent capabilities. But, since all these capabilities, along with general symbolic manipulation, are built-in to Mathematica it can be seen to be an ideal tool for a wide range of analysis of atomic and molecular systems.
REREFENCES 1. R.S. Berry, J. Chem. Ed. 43, 283 (1966) 2. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Reading, 1991) 3. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, New Jersey, 1974) 4. B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules (Longman, UK, 1989) 5. K. Blum, Density Matrix Theory and Applications (Plenum: New York, 1981) 6. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972) 7. J.F. Shriner and W.J. Thompson, Computers in Physics, 7, 144 (1993)
