Basis sets

Vibrational basis set

TROVE provides a number of basis set options for the vibrational basis functions. Currently, all these options are 1D basis sets. With an exception for the Harmonic and Morse bases, all basis set are generated numerically by solving some model 1D Schrödinger equations. The solution methods can vary but are typically based either on the Numerov-Cooley method or variation method. The Numerov-Cooley method is a general shooting type method capable potentially dealing with 1D potentials of any complexity. The variational methods are used in conjunction with different basis sets (FBR) or even with DVR. in the following, the general basis set input is introduced with different basis options explained in detail.

Basis set block

The structure of the Basis set block in the TROVE input is best illustrated using a specific example. Here we use the XY:sub:’2’ input as a typical example, such as SiH₂:

BASIS
 0,'JKtau', Jrot 0
 1,'numerov','linear', 'morse',  range 0, 12, r 8, weight 2.0, points 1000,borders -0.8,1.40
 1,'numerov','linear', 'morse',  range 0, 12, r 8, weight 2.0, points 1000,borders -0.8,1.40
 2,'numerov','linear', 'linear', range 0, 24, r 8, weight 1.0, points 1000,borders 10.0,160.0 deg
END

1. Sub-groups

Basis functions are combined into sub-groups by their symmetry equivalence and labeled with using an integer label, with the vibrational modes numbers with 1,2,3,… and the rotational basis is referenced to as 0.

The first line (mode) is for the rotational basis set functions and used specify the value of angular momentum $J$ and also in some cases the coverage of the rotational quantum number $K$ . In this entry, JKtau is the type of the rotational basis set indicating the Wang symmetrisation of the type $(-1)^{J+K+\tau}$ , see [23Yurchenko], which is currently the only option in TROVE. For $J>0$ calculations the value of Jrot is changed to $J$ of interest.

Other cards of the mode 0 include KMAX (Lmax) defining the maximal value of the rotational QN $k$ . It is used for the quasi-linear triatomics where the rotational QN $k$ is constrained to the vibrational angular momentum quantum number $l$ as $k=l$ . Therefore, this card effectively defines the range of the vibrational angular momentum for quasi-linear triatomics, $l_{\rm max}$ .

SiH₂ has $3N - 6 = 3$ internal degrees of freedom and thus 3 vibrational basis functions are required. For this example, the first group 1 combines the two stretches SiH` ₁ and SiH₂ based on their symmetry equivalence: these two degrees of freedom transform through each other when acted upon with the C_3v (M) symmetry operations. The sub-group ‘2’ is for the bending mode H-Si-H, which is a stand-alone degree of freedom that transforms independently from the sub-group 1. The grouping is used for producing symmetric combinations of basis functions and only coordinates symmetrically related should be grouped together. Details of this procedure are discussed in Theory and in [17YuYaOv].

2. Basis set method/type

For a given vibrational basis function row, the options are as follows. The first card specifies what the one-dimensional basis functions are. In this example they are numerically generated using the Numerov-Cooley method (Numerov card).

Basis set options

Numerov: basis functions are generated numerically using the Numerov-Cooley method.

harmonic: Analytic matrix elements or numerical values of the harmonic oscillator wavefunctions are used.

morse: basis functions and the corresponding matrix elements are based on the analytic expressions for the Morse polynomials.

Fourier: this basis set is introduced for modes with periodicity in periodic potential functions.

laguerre-k: is used in the direct connection with exact KEOs of triatomics, see sections about Frames.

sinrho-laguerre-k: is another basis set constructed for specific exact KEOs of triatomics, see sections about Frames.

3. Coordinates used to expand KEO

This card specifies how the kinetic energy operator is expanded. For the general linearised KEO, the default option is linear which refers to the expansion in terms of the displacements $\xi_i-\xi_i^{\rm eq}$ . For exact KEOs, which are nor general and different for different systems, frames, coordinates, this card defining the expansion types is usually associated with the KEO used.

Here are the currently available options in TROVE:

‘Rational’: used for triatomic exact KEOs, which corresponds to the expansion of the stretching modes in terms of $\frac{1}{r_i}$ (1st “order”), $\frac{1}{r_i}^2$ (2nd “order”) and $\frac{1}{r_i r_j}$ (mixed term).

Automatic: the KEO constructed as part of the input (under construction).

Bond-Length: is similar to rational but independent for polyatomic molecules together with the Angle type. It has been used for the H₂CS molecule [23MeOwTe].

Angle type is to expand the KEO in terms of the trigonometric combinations $\cos(\alpha)$ (1st order), $1/\tan(\alpha)$ (2nd), $1/\sin(\alpha)$ (3rd), $\sin(\alpha)$ (4th), $1/\tan(\alpha)^2$ and $1/\sin(\alpha)^2$ , where $\alpha$ is a generic angle.

Dihedral is a part of the bond-length/angle/dihedral option; it is used to introduce the following terms in the KEO: $\cos(\delta/2)$ (1st), $\sin(\delta/2)$ (2nd), $\cos(\delta/2)^2$ (3rd), $\cos(\delta/2)\sin(\delta/2)$ (4th) and $\sin(\delta/2)^2$ (5th), where $\delta$ is a generic dihedral mode.

4. Coordinates used to expand PEF

The next card gives the expansion coordinates for the potential. The available options are

linear, used for the displacement $\xi_i-\xi_i^{\rm eq}$ ; a popular option for bending degrees of freedom.

morse, used to represent the expansion in the form of the Morse coordinate $1 - e^{-a (r-r_e)}$ and a good option for the stretching coordinates.

cos(x) is to expand PEF in terms of $\cos\alpha$ , but can lead to singularities at around $\alpha=0$ and $\alpha=\pi$ .

5. Quantum number range

The numbers after range specify the range of vibrational quantum numbers of the one-dimensional functions to be used. For the example here, 0-12 is used for stretches and 0-24 for bends. The range should be consistent with the definition of the maximum polyad number used:

$P_{\rm max} = \sum_i a_i v_i^{\rm max} \le n.$

where $a_i$ are the polyd coefficients (weights), defined in the next card.

The borders card can be combined with the units cards, deg, Degree, Degrees, Bohr, for non-default units, e.g.

2,'laguerre-k','linear','linear', range 0,24, weight 1.0, points 2000, borders  0.,120.0 deg

6. Polyad weights

The number after weight (aka resc) gives the weighting $a_i$ of the vibrational quantum number for that coordinate in $P_{\rm max}$ . Since the Si-H stretches here have a waiting of 2, it only makes sense to generate them from 0-12 if the polyad number is set to 24. The legacy aliases for weight are resc (resonance coefficients).

7-8. Integration points and borders

points and borders specify the number of points and the starting points for the Numerov-Cooley integration as the primary usage. Generating these one-dimensional functions is fast and so many points should be taken. The borders should be set far enough into the classically forbidden region of the potential such that the results are not sensitive to slightly larger or lower values. The units for borders are the same as those used that the potential was expanded in (Morse for stretches and angles in radians for bends in this example). For the Numerov-Cooley method, TROVE will check the numerical wavefunctions for their orthogonality and normalisation. If the latter properties are broken, TROVE will stop and suggest to increase the integration borders.

The second use of the coordinate grids defined by these tow cards is in the symmetrisation sampling procedure. Therefore these cards must be defined even for non Numerov-Cooley integration method.

The details of the primitive basis sets are given in the TROVE output file and will be discussed in output.

Other non-standard options

Reduced (alias r): this card allows to reduce the expansion order of PEF when used to generate the basis set. It is sometimes more efficient for symmetry purposes to use a quadratic-type expansion in place of the full expansion with the order defined by PotOrder.

Periodic indicates that the potential is periodic and defines the periodicity. This property can be used to integrate the 1D problem on a smaller range and then extend by applying the periodic boundary conditions. Example:
5,'fourier','linear', 'linear', range 0,17,  weight  1.0, points  500, borders,0.d0,720.d0 deg, periodic 2
Lvib (Vib_Momentum) is used for systems where the basis set is constructed by diagonalising the vibrational angular momentum $\hat{l}^2$ . The advantage of this construction scheme is that the basis set functions are assigned the vibrational angular momentum value $l$ and associated symmetry. This option is extensively used for the linear molecule C₂H₂, which is classified by irreps of D_nh(M), e.g. $E_{l}$ , where $l$ is the vibrational angular momentum value.

As another example, it an be used to for spherical tops such as ammonia or phosphine to assign the vibrational basis and eignefunctions with he vibrational index $l$ . Since typical basis sets used for these systems are 1D, they do not have this useful property and the lvib option could help recover it.

Postprocess (post): this option is used to postprocess the contracted vibrational basis set generated on a reduced potential or Hamiltonian for the full PEF. It helps improve the basis set by re-optimising it. For example, for the lvib-constructed contracted basis functions, i.e. generated as eigenfunctions of $\hat{l}^2$ , they can be post-processed by eigen-solving a reduced Hamiltonian to obtain a more efficient basis and keep $l$ as a quantum number. Example (from C₂H₂):

BASIS
 0,'JKtau', jrot    0
 1,'numerov','linear',  'morse', range 0, 4, weight 2.0, points 2000, borders -0.3,0.6
 2,'numerov','linear',  'morse', range 0, 3, weight 1.0, points 1000, borders -0.5,0.75
 2,'numerov','linear',  'morse', range 0, 3, weight 1.0, points 1000, borders -0.5,0.75
 3,'harmonic','linear', 'linear',range 0, 6,r 2, weight 1.0, points 2000, borders -1.8,1.8  lvib post
 3,'harmonic','linear', 'linear',range 0, 6,r 2, weight 1.0, points 2000, borders -1.8,1.8  lvib post
 3,'harmonic','linear', 'linear',range 0, 6,r 2, weight 1.0, points 2000, borders -1.8,1.8  lvib post
 3,'harmonic','linear', 'linear',range 0, 6,r 2, weight 1.0, points 2000, borders -1.8,1.8  lvib post
END

Here, the harmonic basis set was used for the sub-group 4 combing four linearised bending degrees of freedom of C₂H₂ as the basis for eigen-solving for the vibrational angular momentum $\hat{l}^2$ (lvib). After the new wavefunctions are obtained as classified by $l$ , they are re-optimised (post) for the given $l$ by solving an eigenvalue problem for a reduced 4D Hamiltonian with a quadratic PEF (r 2).

Nocheck is used to suppress checking of the symmetry equivalence of the modes within the same sub-group. This is necessary for the modes which are dynamically symmetry equivalent. For example, when treating molecule CH₃OH can be treat a C:subs:`3v`(M) molecule, the individual stretching CH modes are not equivalent at any fixed torsional configuration and would not be allowed in TROVE to be used for generating the basis sets. Instead, TROVE would choose the 1st mode at some reference torsional angle to generate a reference basis set and will used it for all three modes. For example:
3, 'numerov', 'linear', 'morse', range  0, 4 , weight 1.0,points 1000,borders -0.4,  2.23 nocheck
3, 'numerov', 'linear', 'morse', range  0, 4 , weight 1.0,points 1000,borders -0.4,  2.23 nocheck
3, 'numerov', 'linear', 'morse', range  0, 4 , weight 1.0,points 1000,borders -0.4,  2.23 nocheck

Examples of Basis

H₂O

BASIS
  0,'JKtau', Jrot 0, krot  4
  1,'numerov','rational', 'morse',  range 0,12, r 8, weight 2.0, points  1000, borders -0.36,1.4
  1,'numerov','rational', 'morse',  range 0,12, r 8, weight 2.0, points  1000, borders -0.36,1.4
  2,'laguerre-k','linear','linear', range 0,24,      weight 1.0, points 2000, borders  0.,120.0 deg
END

NH₃

BASIS
 0,'JKtau', Jrot 2
 1,'numerov','linear',  'morse',  range 0, 4, r 8, weight 4.0, points 2000, borders -0.4,2.0
 1,'numerov','linear',  'morse',  range 0, 4, r 8, weight 4.0, points 2000, borders -0.4,2.0
 1,'numerov','linear',  'morse',  range 0, 4, r 8, weight 4.0, points 2000, borders -0.4,2.0
 2,'harmonic','linear', 'linear', range 0,12, r 2, weight 2.0, points 9000, borders -1.90,1.91
 2,'harmonic','linear', 'linear', range 0,12, r 2, weight 2.0, points 9000, borders -1.90,1.92
 3,'numerov','linear',  'linear', range 0,12, r 8, weight 1.0, points 1000, borders -55.0, 55.0 deg
END

CH₄

BASIS
   0,'JKtau', Jrot 0
   1,'numerov','linear',  'morse', r 8, range 0, 0, weight 2.0, points 1000, borders -0.45,0.9
   2,'numerov','linear',  'morse', r 8, range 0, 0, weight 2.0, points 1000, borders -0.45,0.9
   2,'numerov','linear',  'morse', r 8, range 0, 0, weight 2.0, points 1000, borders -0.45,0.9
   2,'numerov','linear',  'morse', r 8, range 0, 0, weight 2.0, points 1000, borders -0.45,0.9
   3,'numerov','linear',  'linear',r 8, range 0, 0, weight 1.0, points 1000, borders -2.10,2.10 post
   3,'numerov','linear',  'linear',r 8, range 0, 0, weight 1.0, points 1000, borders -2.10,2.10 post
   3,'numerov','linear',  'linear',r 8, range 0, 0, weight 1.0, points 1000, borders -2.10,2.10 post
   4,'harmonic','linear', 'linear',r 2, range 0, 6, weight 1.0, points 4000, borders -2.20,2.20 post
   4,'harmonic','linear', 'linear',r 2, range 0, 6, weight 1.0, points 4000, borders -2.20,2.20 post
END

Rotational basis set

For the rotational motion, the standard rigid rotor wavefunctions are used, which are then symmetrised via the Wang-combinations (all expect for some special cases of the Td symmetry) in the $J-K-\tau$ form as follows

$\begin{split} &|J,0,\tau\rangle = |J,0\rangle, \quad \tau = J\; {\rm mod}\; 2 , \\ &|J,K,\tau=0\rangle = \frac{1}{\sqrt{2}} \left[ |J,K,m\rangle + (-1)^{J+K} |J,-K,m\rangle \right],\\ &|J,K,\tau=1\rangle = \frac{i (-1)^{\sigma} }{\sqrt{2}} \left[ |J,K,m\rangle - (-1)^{J+K} |J,-K,m\rangle \right]. \end{split}$

Here $K=|k|$ , $\tau_{\rm rot}$ is the value associated with the parity of

System Message: WARNING/2 (\ket{J,K,\tau_{\rm rot}})

latex exited with error [stdout] This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2021-11-15> patch level 1 L3 programming layer <2022-01-21> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2021/10/04 v1.4n Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-dvips.def) No file math.aux. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \ket l.13 \fontsize{12}{14}\selectfont $\ket {J,K,\tau_{\rm rot}}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 304 bytes). Transcript written on math.log.

and $m$ is omitted on the left-hand side for simplicity’s sake. $K$ is the rotational quantum number ( $K_a$ or $K_c$ , depending on the orientation of the $z$ axis). The sign of $k$ is, however, not a physically meaningful quantity of a rotational eigen-state, but the parity $\tau_{\rm rot}$ is. $\sigma_{\rm rot} = K\, {\rm mod}\, 3$ . The symmetry classification must be implemented for each molecule/symmetry case as part of the subroutines ML_rotsymmetry_<MOLECULE> for each nodule mol_<MOLECULE>.f90. For example, such an implementation for the rotational basis functions $|J,K,\tau\rangle$ of C_3v(M) are given by

$K$	$\tau$	$\Gamma_{\rm rot}$
$3n$	0	$A_1$
$3n$	1	$A_2$
$3n\pm 1$	0	$E_a$
$3n\pm 1$	1	$E_b$

Once implemented, there is nothing else to be specified in the input file as far as the rotational basis set is concerned, except to set the value of $J$ :

BASIS
   0,'JKtau', Jrot 15
.....

For a Td symmetry molecule like methane or silane, the symmetry-adapted rotational functions are obtained as linear combinations of

System Message: WARNING/2 (\ket{J,k,m})

latex exited with error [stdout] This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2021-11-15> patch level 1 L3 programming layer <2022-01-21> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2021/10/04 v1.4n Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-dvips.def) No file math.aux. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \ket l.13 \fontsize{12}{14}\selectfont $\ket {J,k,m}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 256 bytes). Transcript written on math.log.

with $k=-J\ldots J$ , spanning multiple values of $k$ . Therefore, the rotational quantum number $K$ can no longer be used for classification of these symmetrised rigid-rotor combinations. Instead they can be labelled as

System Message: WARNING/2 (\ket{J,\Gamma,n})

latex exited with error [stdout] This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2021-11-15> patch level 1 L3 programming layer <2022-01-21> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2021/10/04 v1.4n Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-dvips.def) No file math.aux. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \ket l.13 \fontsize{12}{14}\selectfont $\ket {J,\Gamma,n}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 296 bytes). Transcript written on math.log.

, where $\Gamma$ is the symmetry and $n$ is a counting index:

$|J,\Gamma,n\rangle = \sum_{k} T_{n,k}^{(J,\Gamma)} |J,k,m\rangle.$

The coefficients $T_{n,k}^{(J,\Gamma)}$ are generated based on the symmetry properties of the transformations of the Euler angles in $|J,k,m\rangle$ . All expansion coefficients required for this symmetry adaptation are generated automatically as part a numerical procedure. For this type of molecules, it is necessary to switch on by placing the card rotsym euler inside the CONTRACTION block, e.g.

CONTRACTION
   Npolyads      14
   rotsym        euler
   model j=0
END