lib_bound: bound state calculations

Potential energy matrix elements

Potential matrix elements: atom + open-shell diatom, body fixed

qmsV = qms_V_bf_DLMK2(sjmkf, sjmki, sLMK, qnames)

Compute matrix elements of the complex conjugate of a Wigner rotation matrix in a symmetric top basis.

\[\begin{split}\begin{align} V_{fi}(L_n,M_n,K_n) & = \langle j_f m_f k_f | D^{(L)\ast}_{MK}|j_i m_i k_i\rangle\\ &= \frac{[j_f,j_i]^{1/2}}{8\pi^2} \int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta \int_0^{2\pi} d\gamma\\ &~~~~\mbox{}\times D^{(j_f)}_{m_fk_f}(\alpha,\beta,\gamma) D^{(L_n)\ast}_{M_nK_n}(\alpha,\beta,\gamma) D^{(j_i)\ast}_{m_ik_i}(\alpha,\beta,\gamma)\,\\ &= [j_f,j_i]^{1/2} (-1)^{m_f-k_f} \begin{pmatrix} j_f & L_n & j_i\\ -m_f & M_n & m_i \end{pmatrix} \begin{pmatrix} j_f & L_n & j_i\\ -k_f & K_n & k_i \end{pmatrix}, \end{align}\end{split}\]

where \([j_f,f_i] = (2j_f+1)(2j_i+1)\) and \(D^{(L)}_{MK}\) are Wigner D-matrices.

This function implements Eqs. (11) and (19) in a study of He+HF ⁺ [DZGvdA04]. It can also be used for Eq. (20) in a study of He+CO ₂ [SvdAG23]. Note that a factor \(\sqrt{(2j'+1)(2j+1)}\) is missing in that equation.

Input arguments

sjmkf:

A q_set data structure (see lib_qmat) containing the set of bra quantum numbers: \(\{j_f m_f k_f; f=1,2,\ldots\}\)

Or:

A Scilab matrix with two columns, with sjmkf(f,:) = [j_f m_f k_f]

sjmki:

A q_set data structure (see lib_qmat) containing the set of ket quantum numbers \(\{j_i m_i k_i; i=1,2,\ldots\}\)

Or:

A Scilab matrix with two columns, with sjmki(i,:) = [j_i m_i k_i]

sLMK:

A q_set data structure (see lib_qmat) containing \(LMK\) of the Wigner D-matrix element \(D^{(L)\ast}_{MK}\).

Or:

A Scilab \(n \times 3\) matrix Q with elements \(Q(i,:)=[L_i M_i K_i]\)

qnames:

A vector of strings with the names of the quantum numbers. By default qnames = ['j' 'm' 'k' 'L' 'M' 'K']

Output argument

qmsV:

A q_mats data structure (see lib_qmat) containing a list of matrices with Gaunt coefficients for all values of \(L\)

qmsV.s1    = sjmkf
qmsV.s2    = sjmki
qmsV.sa    = sLMK
qmsV.As(i) = V_n   // list of matrices V_n(f, i) = <sjmkf(f,:)|D*L_MK|sjmki(i,:)>

Example:

Here is an example of how to use this function:

// Example usage of the function qms_V_bf_PL2

lib_load(['lib_bound' 'lib_special' 'lib_tenop'])

jmk = [0 0 0
       1 0 0
       1 0 1
       2 0 0
       2 0 1
       2 0 2]

LMKs  = [0 0 0
         1 0 0
         1 0 1
         2 0 0
         2 0 1
         2 0 2]

qmsV1 = qms_V_bf_DLMK2(jmk, jmk, LMKs, ['j' 'm' 'k' 'L' 'M' 'K'])
// |J(3), K(3)>A(6;6)<(same)| [L(5); 5]

// Or, create q_sets first

s1    = q_set(jmk, ['j' 'm' 'k'])
sL    = q_set(LMKs, ['L' 'M' 'K'])       // LMKs has 3 columns
qmsV2 = qms_V_bf_DLMK2(s1, s1, sL, [s1.qnames sL.qnames])

if qmsV1==qmsV2
  mprintf('The same!\n')
else
  mprintf('qmsV1 <> qmsV2\n')
end

Gaunt coefficients

qmsV = qms_V_bf_PL2(sjkf, sjki, sL, qnames)

This function computes matrices with Gaunt coefficients:

\[\begin{split}\begin{align} V_{fi}(L) & = \langle j_f k_f | P_L|j_i k_i\rangle\\ &=\int_0^\pi \! \int_0^{2\pi} Y^\ast_{j_fk_f}(\theta,\phi) P_L(\cos\theta) Y_{j_ik_i}(\theta,\phi)\, d\phi \, \sin\theta d\theta\\ &= [(2j_f+1)(2j_i+1)]^{1/2} (-1)^k \begin{pmatrix} j_f & L & j_i\\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} j_f & L & j_i\\ -k_f & 0 & k_i \end{pmatrix}, \end{align}\end{split}\]

where \(Y_{jk}\) are spherical harmonics with the Condon and Shortley phase convention and \(P_L\) are Legendre polynomials.

Input arguments

sjkf:

A q_set data structure (see lib_qmat) containing the set of bra quantum numbers: \(\{j_f k_f; f=1,2,\ldots\}\)

Or:

A Scilab matrix with two columns, with sjkf(f,:) = [j_f k_f]

sjki:

A q_set data structure (see lib_qmat) containing the set of ket quantum numbers \(\{j_i k_i; i=1,2,\ldots\}\)

Or:

A Scilab matrix with two columns, with sjki(i,:) = [j_i k_i]

sL:

A q_set data structure (see lib_qmat) containing the orders \(L\) or the Legendre polynomials \(P_L\)

Or:

A Scilab vector with the values of \(L\)

qnames:

A vector of strings with the names of the quantum numbers. By default qnames = ['j' 'k' 'L']

Output argument

qmsV:

A q_mats data structure (see lib_qmat) containing a list of matrices with Gaunt coefficients for all values of \(L\)

qmsV.s1    = sjkf
qmsV.s2    = sjki
qmsV.sa    = sL
qmsV.As(i) = VL  // list of Gaunt coefficient matrices for: L = sL(i, 'L')
                 // VL(f, i) = <sjkf(f, :)|P_L|sjki(i,:)>

Example:

Here is an example of how to use this function:

// Example usage of the function qms_V_bf_PL2

lib_load(['lib_bound' 'lib_special' 'lib_tenop'])

jk = [0 0
      1 0
      1 1
      2 0
      2 1
      2 2]

Ls  = 0:4

qmsV1 = qms_V_bf_PL2(jk, jk, Ls, ['j' 'k' 'L'])
// |J(3), K(3)>A(6;6)<(same)| [L(5); 5]

// Or, create q_sets first

s1    = q_set(jk, ['j' 'k'])
sL    = q_set(Ls(:), 'L')        // q_set requires a column vector here
qmsV2 = qms_V_bf_PL2(s1, s1, sL, [s1.qnames 'L'])

if qmsV1==qmsV2
  mprintf('The same!\n')
else
  mprintf('qmsV1 <> qmsV2\n')
end

[DZGvdA04]

G. Dhont, W. B. Zeimen, G. C. Groenenboom, and A. van der Avoird. Theoretical study of the He–HF⁺ complex; ii. rovibronic states from coupled diabatic potential energy surfaces. J. Chem. Phys., 120:103–116, 2004. doi:10.1063/1.1629672.

[SvdAG23]

Taha Selim, Ad van der Avoird, and Gerrit C. Groenenboom. State-to-state rovibrational transition rates for CO₂ in the bend mode in collisions with he atoms. J. Chem. Phys., 159:164310, 2023. arXiv:2309.03781, doi:10.1063/5.0174787.