Maple Quantum Chemistry Toolbox
The Maple Quantum Chemistry Toolbox from RDMChem, a separate addon product to Maple, is a powerful environment for the computation and visualization of the electronic structure of molecules. In Maple 2024, this toolbox has significant new features and enhancements that enable: (1) Computing electronic energies and properties with a new generalization of density functional theory that captures static correlation, (2) Running the CoupledCluster and NuclearGradient commands on Windows that were previously available only on macOS and Linux, (3) Harnessing AI through a new command, Chat, that when prompted with a word or phrase, can provide invaluable information about molecules and compounds, electronic structure methods, as well as other scientific words or concepts, (4) Learning or teaching about the role of chemistry in art through a new curricular set of lessons entitled "The Chemistry of Art," and (5) Experiencing additional enhancements and improvements throughout the package.
Note that the Maple Quantum Chemistry Toolbox (QCT) is required in order to execute the examples in this worksheet.

Generalization of Density Functional Theory


A well known limitation of density functional theory (DFT) is its difficulty in predicting the energies and properties of molecules with static correlation. Static correlation is important to the accurate prediction of charges, van der Waals forces, barrier heights, and bi and multiradicals. Introduced in QCT 2024 is a generalization of DFT that can treat static correlation. The new command, RDMFunctional adds a universal correction, based on the 1electron reduced density matrix (1RDM) rather than the density alone, to any DFT functional.
While the generalized DFT is applicable to a wide range of molecules, below you can see the command with the dissociation of the diatomic molecule N2. First, load the QuantumChemistry package
>

$\mathrm{with}\left(\mathrm{QuantumChemistry}\right)\;$

$\left[{\mathrm{AOLabels}}{\,}{\mathrm{ActiveSpaceCI}}{\,}{\mathrm{ActiveSpaceSCF}}{\,}{\mathrm{AtomicData}}{\,}{\mathrm{BondAngles}}{\,}{\mathrm{BondDistances}}{\,}{\mathrm{Charges}}{\,}{\mathrm{ChargesPlot}}{\,}{\mathrm{Chat}}{\,}{\mathrm{ContractedSchrodinger}}{\,}{\mathrm{CorrelationEnergy}}{\,}{\mathrm{CoupledCluster}}{\,}{\mathrm{DensityFunctional}}{\,}{\mathrm{DensityPlot3D}}{\,}{\mathrm{Dipole}}{\,}{\mathrm{DipolePlot}}{\,}{\mathrm{Energy}}{\,}{\mathrm{ExcitationEnergies}}{\,}{\mathrm{ExcitationSpectra}}{\,}{\mathrm{ExcitationSpectraPlot}}{\,}{\mathrm{ExcitedStateEnergies}}{\,}{\mathrm{ExcitedStateSpins}}{\,}{\mathrm{ExcitonDensityPlot}}{\,}{\mathrm{ExcitonPopulations}}{\,}{\mathrm{ExcitonPopulationsPlot}}{\,}{\mathrm{FullCI}}{\,}{\mathrm{GeometryOptimization}}{\,}{\mathrm{HartreeFock}}{\,}{\mathrm{Interactive}}{\,}{\mathrm{Isotopes}}{\,}{\mathrm{LiteratureSearch}}{\,}{\mathrm{MOCoefficients}}{\,}{\mathrm{MODiagram}}{\,}{\mathrm{MOEnergies}}{\,}{\mathrm{MOIntegrals}}{\,}{\mathrm{MOOccupations}}{\,}{\mathrm{MOOccupationsPlot}}{\,}{\mathrm{MOSymmetries}}{\,}{\mathrm{MP2}}{\,}{\mathrm{MolecularData}}{\,}{\mathrm{MolecularDictionary}}{\,}{\mathrm{MolecularGeometry}}{\,}{\mathrm{NuclearEnergy}}{\,}{\mathrm{NuclearGradient}}{\,}{\mathrm{OscillatorStrengths}}{\,}{\mathrm{Parametric2RDM}}{\,}{\mathrm{PlotMolecule}}{\,}{\mathrm{Populations}}{\,}{\mathrm{Purify2RDM}}{\,}{\mathrm{QuantumComputing}}{\,}{\mathrm{RDM1}}{\,}{\mathrm{RDM2}}{\,}{\mathrm{RDMFunctional}}{\,}{\mathrm{RTM1}}{\,}{\mathrm{ReadXYZ}}{\,}{\mathrm{Restore}}{\,}{\mathrm{Save}}{\,}{\mathrm{SaveXYZ}}{\,}{\mathrm{SearchBasisSets}}{\,}{\mathrm{SearchFunctionals}}{\,}{\mathrm{SkeletalStructure}}{\,}{\mathrm{SolventDatabase}}{\,}{\mathrm{Thermodynamics}}{\,}{\mathrm{TransitionDipolePlot}}{\,}{\mathrm{TransitionDipoles}}{\,}{\mathrm{TransitionOrbitalPlot}}{\,}{\mathrm{TransitionOrbitals}}{\,}{\mathrm{Variational2RDM}}{\,}{\mathrm{VibrationalModeAnimation}}{\,}{\mathrm{VibrationalModes}}{\,}{\mathrm{Video}}\right]$
 (1.1) 
Select a set of bond distances from the roots of the sixthorder Chebyshev polynomial
>

$\mathrm{bond\_distances}\u2254\mathrm{map}\left(x\to xplus;2.0comma;\left[\mathrm{fsolve}\left(\mathrm{expand}\left(\mathrm{ChebyshevT}\left(8comma;x\right)\right)\right)\right]\right)semi;\phantom{\rule[0.0ex]{0.0em}{0.0ex}}$

${\mathrm{bond\_distances}}{\u2254}\left[{1.01921472}{\,}{1.16853039}{\,}{1.44442977}{\,}{1.80490968}{\,}{2.19509032}{\,}{2.55557023}{\,}{2.83146961}{\,}{2.98078528}\right]$
 (1.2) 
and define a list of molecular geometries with each geometry corresponding to one of the bond distances
>

$\mathrm{molecules}\u2254\left[\mathrm{seq}\left(\left[\left[''N''comma;0comma;0comma;0\right]comma;\left[''N''comma;0comma;0comma;R\right]\right]comma;R\mathbf{in}\mathrm{bond\_distances}\right)\right]semi;$

${\mathrm{molecules}}{\u2254}\left[\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{1.01921472}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{1.16853039}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{1.44442977}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{1.80490968}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{2.19509032}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{2.55557023}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{2.83146961}\right]\right]{\,}\left[\left[{''N''}{\,}{0}{\,}{0}{\,}{0}\right]{\,}\left[{''N''}{\,}{0}{\,}{0}{\,}{2.98078528}\right]\right]\right]$
 (1.3) 
The energies for each geometry may be then readily computed from DensityFunctional with the Energy
command in the Quantum Chemistry package,
>

$\mathrm{energies\_dft}\u2254\left[\mathrm{seq}\left(\mathrm{Energy}\left(\mathrm{molecule}comma;\mathrm{method}equals;\mathrm{DensityFunctional}comma;\mathrm{basis}equals;''ccpVDZ''comma;\mathrm{xc}equals;''B3LYP''\right)comma;\mathrm{molecule}\mathbf{in}\mathrm{molecules}\right)\right]semi;$

${\mathrm{energies\_dft}}{\u2254}\left[{\mathrm{109.50786339}}{\,}{\mathrm{109.52331040}}{\,}{\mathrm{109.37588586}}{\,}{\mathrm{109.17741963}}{\,}{\mathrm{109.03358249}}{\,}{\mathrm{108.95146495}}{\,}{\mathrm{108.90997138}}{\,}{\mathrm{108.89291096}}\right]$
 (1.4) 
and using polynomial interpolation, generate a polynomial in the bond distance R
>

$\mathrm{pes\_dft}\u2254\mathrm{interp}\left(\mathrm{bond\_distances}comma;\mathrm{energies\_dft}comma;R\right)semi;$

${\mathrm{pes\_dft}}{\u2254}{}{0.14339035}{{R}}^{{7}}{+}{2.20942302}{{R}}^{{6}}{}{14.44738600}{{R}}^{{5}}{+}{51.96535307}{{R}}^{{4}}{}{110.97891955}{{R}}^{{3}}{+}{140.38786698}{{R}}^{{2}}{}{96.50876625}{R}{}{81.97856683}$
 (1.5) 
Similarly, you can use the new RDMFunctional method in the Energy command
>

$\mathrm{energies\_rdm}\u2254\left[\mathrm{seq}\left(\mathrm{Energy}\left(\mathrm{molecule}comma;\mathrm{method}equals;\mathrm{RDMFunctional}comma;\mathrm{basis}equals;''ccpVDZ''comma;\mathrm{xc}equals;''B3LYP''\right)comma;\mathrm{molecule}\mathbf{in}\mathrm{molecules}\right)\right]semi;$

${\mathrm{energies\_rdm}}{\u2254}\left[{\mathrm{109.50786339}}{\,}{\mathrm{109.52331040}}{\,}{\mathrm{109.37588586}}{\,}{\mathrm{109.18887170}}{\,}{\mathrm{109.10062254}}{\,}{\mathrm{109.06672079}}{\,}{\mathrm{109.05806429}}{\,}{\mathrm{109.05565802}}\right]$
 (1.6) 
and generate a polynomial in R
>

$\mathrm{pes\_rdm}\u2254\mathrm{interp}\left(\mathrm{bond\_distances}comma;\mathrm{energies\_rdm}comma;R\right)semi;$

${\mathrm{pes\_rdm}}{\u2254}{0.00403016}{{R}}^{{7}}{+}{0.13029220}{{R}}^{{6}}{}{2.19462464}{{R}}^{{5}}{+}{12.96198119}{{R}}^{{4}}{}{38.72108443}{{R}}^{{3}}{+}{62.54064536}{{R}}^{{2}}{}{51.32917602}{R}{}{92.88749381}$
 (1.7) 
The potential energy curves from DensityFunctional (red) and RDMFunctional (blue) can be plotted together
>

$\mathrm{p\_dft}\u2254\mathrm{plot}\left(\mathrm{pes\_dft}comma;Requals;1.0..2.9comma;\mathrm{axes}equals;\mathrm{boxed}comma;\mathrm{labels}equals;\left[\'\'Bond\; Distance\; (\xc3\dots )\'\'comma;"Energy\left(\mathrm{hartree}\right)quot;\right]comma;\mathrm{color}equals;\mathrm{red}comma;\mathrm{thickness}equals;3comma;\mathrm{labeldirections}equals;\left[\mathrm{horizontal}comma;\mathrm{vertical}\right]\right)colon;\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\mathrm{p\_rdm}\u2254\mathrm{plot}\left(\mathrm{pes\_rdm}comma;Requals;1.0..2.9comma;\mathrm{axes}equals;\mathrm{boxed}comma;\mathrm{labels}equals;\left[\'\'Bond\; Distance\; (\xc3\dots )\'\'comma;"Energy\left(\mathrm{hartree}\right)quot;\right]comma;\mathrm{color}equals;\mathrm{blue}comma;\mathrm{thickness}equals;3\right)colon;\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\mathrm{plots}:\mathrm{display}\left(\left\{\mathrm{p\_dft}comma;\mathrm{p\_rdm}\right\}\right)semi;$

 
While the DFT dissociation curve (red) rises too quickly after 1.8$\u27e6\AA \u27e7$due to its incorrect treatment of static correlation, the RDMFunctional dissociation curve (blue) exhibits the correct behavior, leveling off as the bond breaks.


Enhanced Windows Edition


The Quantum Chemistry Toolbox has been updated on Windows to include the commands CoupledCluster and NuclearGradient that were previously available only on macOS and Linux. The CoupledCluster provides access to the coupled cluster methods, CCSD and CCSD(T), and NuclearGradient implements analytical gradients which significantly accelerate geometry optimizations.
Consider the molecule 1,3difluorobenzene
>

$\mathrm{mol}\u2254\mathrm{MolecularGeometry}\left(''1,3difluorobenzene''\right)semi;$

${\mathrm{mol}}{\u2254}\left[\left[{''F''}{\,}{2.36790000}{\,}{1.02540000}{\,}{\mathrm{0.00020000}}\right]{\,}\left[{''F''}{\,}{\mathrm{2.36810000}}{\,}{1.02530000}{\,}{0.00010000}\right]{\,}\left[{''C''}{\,}{\mathrm{0.00030000}}{\,}{1.05310000}{\,}{0.00010000}\right]{\,}\left[{''C''}{\,}{1.20790000}{\,}{0.35580000}{\,}{0.00020000}\right]{\,}\left[{''C''}{\,}{\mathrm{1.20810000}}{\,}{0.35540000}{\,}{\mathrm{0.00020000}}\right]{\,}\left[{''C''}{\,}{1.20820000}{\,}{\mathrm{1.03900000}}{\,}{0.00010000}\right]{\,}\left[{''C''}{\,}{\mathrm{1.20790000}}{\,}{\mathrm{1.03940000}}{\,}{\mathrm{0.00010000}}\right]{\,}\left[{''C''}{\,}{0.00030000}{\,}{\mathrm{1.73660000}}{\,}{0}\right]{\,}\left[{''H''}{\,}{\mathrm{0.00040000}}{\,}{2.13890000}{\,}{0.00010000}\right]{\,}\left[{''H''}{\,}{2.14900000}{\,}{\mathrm{1.58160000}}{\,}{0.00010000}\right]{\,}\left[{''H''}{\,}{\mathrm{2.14840000}}{\,}{\mathrm{1.58220000}}{\,}{\mathrm{0.00010000}}\right]{\,}\left[{''H''}{\,}{0.00040000}{\,}{\mathrm{2.82270000}}{\,}{0.00010000}\right]\right]$
 (2.1) 
>

$\mathrm{PlotMolecule}\left(\mathrm{mol}\right)\;$

On any supported platform including Windows you can use the command CoupledCluster to compute its groundstate energy and properties with coupled cluster with single and double excitations${}$(CCSD)
>

$\mathrm{data}\u2254\mathrm{CoupledCluster}\left(\mathrm{mol}\right)semi;$

Similarly, on Windows you can now compute the analytical nuclear gradient for 1,3difluorobenzene in seconds
>

$\mathrm{grad}\u2254\mathrm{NuclearGradient}\left(\mathrm{mol}comma;\mathrm{method}equals;\mathrm{CoupledCluster}\right)semi;$

which substantially accelerates molecular geometry optimization using the GeometryOptimization command.


Power of AI through New Command Chat


QCT 2024 combines a modern quantum chemistry package with the power of generative AI through natural language models. With the new command Chat you can ask AI to define a molecule, drug, or compound or to explain a scientific concept, terminology, or method. (Note that before using Chat, you need to review and agree to the AI Terms of Use.)
For example, you can ask Chat for information about buckeyballs
>

$\mathrm{Chat}\left(''Buckminsterfullerene''\right)\;$

${''Buckminsterfullerene,\; also\; known\; as\; a\; buckyball,\; is\; a\; specific\; type\; of\; molecular\; formation\; that\; includes\; 60\; carbon\; atoms\; arranged\; in\; a\; spherical\; structure,\; similar\; in\; shape\; to\; a\; soccer\; ball.\; It\; was\; named\; after\; Richard\; Buckminster\; Fuller,\; a\; renowned\; architect\; known\; for\; designing\; structures\; similar\; to\; the\; shape\; of\; this\; molecule.\; This\; particular\; molecule\; is\; a\; type\; of\; fullerene,\; a\; category\; of\; carbon\; structures\; which\; also\; includes\; carbon\; nanotubes.\; Buckminsterfullerenes\; are\; significant\; in\; scientific\; research\; due\; to\; their\; unique\; physical\; and\; chemical\; properties.\; They\; are\; used\; in\; a\; variety\; of\; applications,\; ranging\; from\; conductors\; and\; superconductors\; to\; medical\; applications.\; The\; buckminsterfullerene\; was\; discovered\; in\; 1985\; by\; a\; team\; of\; scientists\; including\; Harry\; Kroto,\; Richard\; Smalley,\; and\; Robert\; Curl,\; for\; which\; they\; were\; awarded\; the\; Nobel\; Prize\; in\; Chemistry\; in\; 1996.''}$
 (3.1) 
or the antibiotic drug penicillin
>

$\mathrm{Chat}\left(''penicillin''\right)\;$

 (3.2) 
If you want to know more about penicillin, we can just ask for more
>

$\mathrm{Chat}\left(\mathrm{more}\right)\;$

 (3.3) 
You can also inquire about any scientific concept or terminology; for example, we can ask about entanglement
>

$\mathrm{Chat}\left(''entanglement''\right)\;$

 (3.4) 
or even an electronic structure method like density functional theory
>

$\mathrm{Chat}\left(''density\; functional\; theory''\right)\;$

 (3.5) 
The command Chat allow you to connect to the power of AI without leaving either Maple or the Quantum Chemistry package.


Using the Package in the Classroom


The Maple Quantum Chemistry Toolbox includes approximately 30 lessons that can be used in chemistry and physics courses from advanced high school courses through the graduate level. These lessons and associated curricula provide instructors and students with realtime quantum chemistry computations and visualizations that quickly deepen understanding of molecular concepts. Detailed lesson plans and curricula are provided for Introductory (General) Chemistry, Physical Chemistry (Quantum Mechanics and Thermodynamics), Thermodynamics (Physics), Quantum Mechanics (Physics), Computational Chemistry, and Quantum Chemistry as well as Advanced Placement (AP) and International Baccalaureate (IB) chemistry courses. Topics include atomic structure, chemical bonding, the MaxwellBoltzmann distribution, heat capacity, enthalpy, entropy, free energy, particleinabox, vibrational normal modes, infrared spectroscopy, as well as advanced electronic structure methods. Additional resources and lessons are available at the Great Quantum Chemistry Dictionary and in the Maple Application Center. Use of the QCT in the classroom is described in a recent paper in J. Chem. Ed.
QCT 2024 includes a new curricular set of lessons entitled "The Chemistry of Art." For example, the following sections provide two short excerpts from lessons in which (1) the visual effects of color vision deficiency (CVD) are explored through an interactive embedded component and (2) quantum calculations of the indigo and tyrian purple dyes are performed to compare their absorption spectra.

Color Vision Deficiency (CVD)


Using Maple, you can simulate the effect of CVD on the perception of a sample color palette plotted in the RGB color space. Select a CVD type, and then slowly increase the severity from 0 to 100 and watch as the colorspace is distorted and each color in the palette changes. You may find it useful to rotate the plots as you increase the severity. If you personally experience a form of color blindness, you can still describe the effect of the simulation as the points representing the colors in the gamut contract into a plane. You should also recall what property is plotted along each axis.
>

$\mathrm{colors}\u2254\left[\mathrm{seq}\left(\mathrm{RandomTools}:\mathrm{Generate}\left(\mathrm{list}\left(\mathrm{float}\left(\mathrm{range}=0..1\,\mathrm{digits}=4\,\mathrm{method}=\mathrm{uniform}\right)\,3\right)\right)\,i=1..200\right)\right]\:\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\mathrm{Explore}\left(\mathrm{plots}:\mathrm{display}\left(\mathrm{Vector}\left[\mathrm{row}\right]\left(\left[\mathrm{ColorTools}:\mathrm{SpatterPlot3d}\left(\mathrm{colors}\,\mathrm{symbol}equals;''box''comma;\mathrm{space}equals;''RGB''comma;\mathrm{labelfont}equals;\left[\mathrm{Helvetica}comma;16\right]comma;\mathrm{orientation}equals;\left[20comma;45comma;38\right]\right)comma;\mathrm{ColorTools}:\mathrm{SpatterPlot3d}\left(\mathrm{map}\left(\mathrm{ColorTools}:\mathrm{CVDSimulation}comma;\mathrm{colors}comma;\mathrm{cvdtype}comma;\mathrm{severity}\right)comma;\mathrm{symbol}equals;''box''comma;\mathrm{space}equals;''RGB''comma;\mathrm{labelfont}equals;\left[\mathrm{Helvetica}comma;16\right]comma;\mathrm{orientation}equals;\left[20comma;45comma;38\right]\right)\right]\right)\right)comma;\mathrm{severity}equals;0..100comma;\mathrm{parameters}equals;\left[\mathrm{cvdtype}equals;\left[''deuteranomaly''comma;''protanomaly''comma;''tritanomaly''\right]\right]comma;\mathrm{placement}equals;\mathrm{bottom}comma;\mathrm{size}equals;\left[1000comma;350\right]\right)semi;$



$\mathbf{cvdtype}$



$\mathbf{severity}$
















Quantum Calculations of Indigo and Tyrian Purple Dyes


Consider two related dyes: indigo and tyrian purple:
>

$\mathrm{indigo}\u2254\left[\left[''C''comma;\mathrm{2.82265000}comma;0.73898000comma;0.00005000\right]comma;\left[''C''comma;\mathrm{3.96802000}comma;1.53537000comma;0.\right]comma;\left[''C''comma;\mathrm{5.20393000}comma;0.88612000comma;\mathrm{0.00005000}\right]comma;\left[''C''comma;\mathrm{5.31138000}comma;\mathrm{0.51504000}comma;\mathrm{0.00005000}\right]comma;\left[''C''comma;\mathrm{4.16296000}comma;\mathrm{1.30350000}comma;\mathrm{0.00001000}\right]comma;\left[''C''comma;\mathrm{2.91836000}comma;\mathrm{0.67319000}comma;0.00003000\right]comma;\left[''H''comma;\mathrm{6.29247000}comma;\mathrm{0.97833000}comma;\mathrm{0.00009000}\right]comma;\left[''H''comma;\mathrm{4.21764000}comma;\mathrm{2.38780000}comma;\mathrm{0.00001000}\right]comma;\left[''H''comma;\mathrm{3.90356000}comma;2.61878000comma;0.00001000\right]comma;\left[''H''comma;\mathrm{6.10933000}comma;1.48632000comma;\mathrm{0.00009000}\right]comma;\left[''C''comma;\mathrm{1.55139000}comma;\mathrm{1.21054000}comma;0.00005000\right]comma;\left[''C''comma;\mathrm{0.68036000}comma;0.00041000comma;0.00006000\right]comma;\left[''N''comma;\mathrm{1.48727000}comma;1.11979000comma;0.00017000\right]comma;\left[''O''comma;\mathrm{1.14282000}comma;\mathrm{2.37426000}comma;0.00004000\right]comma;\left[''H''comma;\mathrm{1.10083000}comma;2.05448000comma;\mathrm{0.00008000}\right]comma;\left[''C''comma;0.68036000comma;\mathrm{0.00040000}comma;\mathrm{0.00006000}\right]comma;\left[''N''comma;1.48727000comma;\mathrm{1.11979000}comma;\mathrm{0.00017000}\right]comma;\left[''C''comma;2.82264000comma;\mathrm{0.73898000}comma;\mathrm{0.00005000}\right]comma;\left[''C''comma;2.91836000comma;0.67319000comma;\mathrm{0.00003000}\right]comma;\left[''C''comma;1.55140000comma;1.21054000comma;\mathrm{0.00004000}\right]comma;\left[''H''comma;1.10081000comma;\mathrm{2.05448000}comma;0.00011000\right]comma;\left[''O''comma;1.14282000comma;2.37426000comma;\mathrm{0.00004000}\right]comma;\left[''C''comma;3.96802000comma;\mathrm{1.53537000}comma;\mathrm{0.}\right]comma;\left[''C''comma;4.16297000comma;1.30350000comma;\mathrm{0.}\right]comma;\left[''C''comma;5.31138000comma;0.51504000comma;0.00004000\right]comma;\left[''C''comma;5.20393000comma;\mathrm{0.88612000}comma;0.00005000\right]comma;\left[''H''comma;6.10932000comma;\mathrm{1.48633000}comma;0.00009000\right]comma;\left[''H''comma;3.90355000comma;\mathrm{2.61878000}comma;0.\right]comma;\left[''H''comma;4.21765000comma;2.38780000comma;0.\right]comma;\left[''H''comma;6.29247000comma;0.97832000comma;0.00007000\right]\right]colon;$

>

$\mathrm{tyrianpurple}\u2254\left[\left[''C''\,\mathrm{7.53629000}\,2.44628000\,0.02964000\right]\,\left[''C''\,\mathrm{6.37087000}\,3.18191000\,0.04692000\right]\,\left[''C''\,\mathrm{5.15824000}\,2.48164000\,0.00715000\right]\,\left[''C''\,\mathrm{5.11665000}\,1.07281000\,\mathrm{0.02689000}\right]\,\left[''C''\,\mathrm{6.29914000}\,0.33034000\,\mathrm{0.01590000}\right]\,\left[''C''\,\mathrm{7.48107000}\,1.04288000\,0.02018000\right]\,\left[''H''\,\mathrm{4.16101000}\,0.55333000\,\mathrm{0.05401000}\right]\,\left[''H''\,\mathrm{6.29526000}\,\mathrm{0.75397000}\,\mathrm{0.03064000}\right]\,\left[''H''\,\mathrm{6.39556000}\,4.26515000\,0.09313000\right]\,\left[''Br''\,\mathrm{3.53176000}\,3.45097000\,0.01113000\right]\,\left[''C''\,\mathrm{8.87511000}\,0.56228000\,0.04867000\right]\,\left[''C''\,\mathrm{9.70911000}\,1.82064000\,0.05880000\right]\,\left[''N''\,\mathrm{8.84729000}\,2.89698000\,0.13866000\right]\,\left[''O''\,\mathrm{9.25455000}\,\mathrm{0.59150000}\,0.06050000\right]\,\left[''H''\,\mathrm{9.17649000}\,3.82873000\,\mathrm{0.08865000}\right]\,\left[''C''\,\mathrm{11.03865000}\,1.75648000\,\mathrm{0.03735000}\right]\,\left[''N''\,\mathrm{11.90080000}\,0.68029000\,\mathrm{0.11662000}\right]\,\left[''C''\,\mathrm{13.21148000}\,1.13137000\,\mathrm{0.00440000}\right]\,\left[''C''\,\mathrm{13.26634000}\,2.53479000\,0.00535000\right]\,\left[''C''\,\mathrm{11.87231000}\,3.01505000\,\mathrm{0.02598000}\right]\,\left[''H''\,\mathrm{11.57160000}\,\mathrm{0.25121000}\,0.11179000\right]\,\left[''O''\,\mathrm{11.49263000}\,4.16874000\,\mathrm{0.03827000}\right]\,\left[''C''\,\mathrm{14.37716000}\,0.39607000\,\mathrm{0.01880000}\right]\,\left[''C''\,\mathrm{14.44796000}\,3.24766000\,0.04478000\right]\,\left[''C''\,\mathrm{15.63061000}\,2.50553000\,0.05891000\right]\,\left[''C''\,\mathrm{15.58949000}\,1.09670000\,0.02446000\right]\,\left[''Br''\,\mathrm{17.21626000}\,0.12784000\,0.02504000\right]\,\left[''H''\,\mathrm{14.35292000}\,\mathrm{0.68716000}\,\mathrm{0.06523000}\right]\,\left[''H''\,\mathrm{14.45146000}\,4.33197000\,0.05995000\right]\,\left[''H''\,\mathrm{16.58602000}\,3.02527000\,0.08893000\right]\right]\:$

>

$\mathrm{plots}:\mathrm{display}\left(\mathrm{Vector}\left[\mathrm{row}\right]\left(\left[\mathrm{PlotMolecule}\left(\mathrm{indigo}\right)\,\mathrm{PlotMolecule}\left(\mathrm{tyrianpurple}\right)\right]\right)\right)\;$

Figure: Structures of indigo (left) and tyrian purple (right).
The only difference between the two structures is that tyrian purple has bromines at the 6 and 6' positions. This does not affect the number of πelectrons or the length of the conjugated chain, so the particleinabox model would treat these two dyes as being identical!
For more complex compounds, one must use more sophisticated methods to calculate the energy levels. Here we use the Quantum Chemistry Toolbox to calculate the ground and excited states of each of these dyes.
After performing DFT and TDDFT calculations for both molecules in the lesson (see the lesson for the explicit calculations), you obtain the following two tables of excitation spectra with the ExcitationSpectra command:
Indigo
spectra_indigo_b3lyp := ExcitationSpectra(indigo, method = DensityFunctional, basis = "631g", nstates = [3, 3], showtable);
State

Energy

Wavelength

Spin

Oscillator

$1$

$1.03972808\u27e6\mathrm{eV}\u27e7$

$1192.46753221\u27e6\mathrm{nm}\u27e7$

Triplet

$0.44895520$

$2$

$2.32548387\u27e6\mathrm{eV}\u27e7$

$533.15440626\u27e6\mathrm{nm}\u27e7$

Singlet

$0.28403090$

$3$

$2.35743686\u27e6\mathrm{eV}\u27e7$

$525.92796749\u27e6\mathrm{nm}\u27e7$

Triplet

$0.08185276$

$4$

$2.42658941\u27e6\mathrm{eV}\u27e7$

$510.94015624\u27e6\mathrm{nm}\u27e7$

Triplet

$4.81491290\times {10}^{\mathrm{10}}$

$5$

$2.79208571\u27e6\mathrm{eV}\u27e7$

$444.05584358\u27e6\mathrm{nm}\u27e7$

Singlet

$2.51377265\times {10}^{\mathrm{10}}$

$6$

$3.46710287\u27e6\mathrm{eV}\u27e7$

$357.60172717\u27e6\mathrm{nm}\u27e7$

Singlet${}$

$0.00641236$



and
Tyrian Purple
spectra_tyrianpurple_b3lyp := ExcitationSpectra(tyrianpurple, method = DensityFunctional, basis = "631g", nstates = [3, 3], showtable);
State

Energy

Wavelength

Spin

Oscillator

$1$

$1.18482603\u27e6\mathrm{eV}\u27e7$

$1046.43377112\u27e6\mathrm{nm}\u27e7$

Singlet

$0.01063373$

$2$

$1.46300077\u27e6\mathrm{eV}\u27e7$

$847.46501988\u27e6\mathrm{nm}\u27e7$

Triplet

$0.42879167$

$3$

$2.46467321\u27e6\mathrm{eV}\u27e7$

$503.04517781\u27e6\mathrm{nm}\u27e7$

Triplet

$5.64622357\times {10}^{\mathrm{7}}$

$4$

$2.59177389\u27e6\mathrm{eV}\u27e7$

$478.37582495\u27e6\mathrm{nm}\u27e7$

Singlet

$0.30183798$

$5$

$2.93685950\u27e6\mathrm{eV}\u27e7$

$422.16591357\u27e6\mathrm{nm}\u27e7$

Triplet

$1.77823855$

$6$

$2.94297131\u27e6\mathrm{eV}\u27e7$

$421.28918132\u27e6\mathrm{nm}\u27e7$

Singlet

$0.00001806$



Looking for the singlet excitations with the largest oscillator strengths, you can predict transitions at 533 nm and 478 nm for indigo and tyrian purple with the relative ordering in reasonable agreement with experimental measurements. Their differences in absorption are responsible for their differences in color!


${}$
