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High-precision calculations for one- and two-valence atomic systems
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High-precision calculations for one- and two-valence atomic systems
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Second-order amplitudes for a valence state/transition: polarisabilities, PNC, etc.
The secondOrder module (run as Module::secondOrder{}) calculates the second-order (in the external field) amplitude \( A^K \) for a single-valence transition \( A \to B \), for two one-body operators \( t \) (at frequency \( \omega \)) and \( s \) (at \( \omega_s = E_B - E_A - \omega \)):
\[ A^K = \sum_n \left[ c_1 \frac{\langle B||t||n\rangle\langle n||s||A\rangle}{E_A+\omega_s-E_n} + c_2 \frac{\langle B||s||n\rangle\langle n||t||A\rangle}{E_A+\omega-E_n} \right]. \]
This single module covers (with the appropriate operators and rank K):
and the relevant physical quantity is printed automatically.
The valence sum is evaluated either by sum-over-states (method = SOS;, over the spectrum or basis), or with mixed states (method = MS;, solving the inhomogeneous equation), which is complete: no truncation of the sum. Run twice to compare the two. The mixed-states method evaluates the sums two independent ways (with the mixed states of s, and of t), which must agree: a strong internal check, printed with their relative difference.
The core polarisation contribution is included (for diagonal, K = 0 amplitudes), and the valence sum is decomposed into core-valence, main (low-n, see n_main), and tail parts, for comparison with other calculations. Structure radiation + normalisation may be added to the matrix elements of the sum via the StructureRadiation{} block (sum-over-states method only).
This is the single-valence analogue of the CI_secondOrder module.
Same definitions in the CI_secondOrder module, with \( j \to J \). \( t \) has rank \( k_t \), \( s \) has rank \( k_s \), and \( [K] \equiv 2K+1 \).
Uncoupled (definite projections \( q_1 \) of \( t \), \( q_2 \) of \( s \); the sum over \( n \) includes magnetic quantum numbers):
\[ A^{k_tk_s}_{q_1q_2} = \sum_n \left[ \frac{\langle B|t_{q_1}|n\rangle\langle n|s_{q_2}|A\rangle} {E_A+\omega_s-E_n} + \frac{\langle B|s_{q_2}|n\rangle\langle n|t_{q_1}|A\rangle} {E_A+\omega-E_n} \right]. \]
Coupled to rank \( K \), with \( Q = q_1+q_2 = m_B-m_A \):
\[ A^K_Q = \sum_{q_1q_2}\langle k_tq_1\,k_sq_2|KQ\rangle\,A^{k_tk_s}_{q_1q_2} = (-1)^{k_t-k_s+Q}\sqrt{[K]}\sum_{q_1q_2} \begin{pmatrix} k_t & k_s & K \\ q_1 & q_2 & -Q \end{pmatrix} A^{k_tk_s}_{q_1q_2}. \]
Reduced, via the Wigner-Eckart theorem:
\[ A^K_Q = (-1)^{j_B-m_B} \begin{pmatrix} j_B & K & j_A \\ -m_B & Q & m_A \end{pmatrix} A^K , \]
\( A^K \) being the quantity the module reports: the reduced matrix element of \( [t\times s]^K \) (both time orderings), with
\[ c_1 = (-1)^{K}\sqrt{[K]}\,(-1)^{j_B+j_A} \begin{Bmatrix} K & k_s & k_t \\ j_n & j_B & j_A \end{Bmatrix}, \qquad c_2 = (-1)^{k_t+k_s}\sqrt{[K]}\,(-1)^{j_B+j_A} \begin{Bmatrix} K & k_t & k_s \\ j_n & j_B & j_A \end{Bmatrix}. \]
Uncoupling again gives the z-component ( \( m_A = m_B = m \), \( q_1 = q_2 = 0 \)), also printed:
\[ A_{zz} = \sum_K \langle k_t0\,k_s0|K0\rangle\,(-1)^{j_B-m} \begin{pmatrix} j_B & K & j_A \\ -m & 0 & m \end{pmatrix} A^K . \]
Sign convention. The coupling above is the standard Clebsch-Gordan one, \( t \) first. The alternative definition
\[ \tilde A^K_Q = (-1)^{Q}\sqrt{[K]}\sum_{q_1q_2} \begin{pmatrix} k_t & k_s & K \\ -q_1 & -q_2 & Q \end{pmatrix} A^{k_tk_s}_{q_1q_2} = (-1)^K A^K_Q \]
differs by \( (-1)^K \). It cancels in \( A_{zz} \) (so in \( E_{\rm PNC} \)) and for even \( K \); it affects only the sign of \( \beta \).
With \( t = s = d \) (E1) and \( [j] \equiv 2j+1 \):
\[ \alpha_0 = \frac{A^0}{\sqrt{3[j_A]}} \quad (K=0,\ B=A), \qquad \alpha_2 = -\sqrt{\frac{2j(2j-1)}{3(j+1)(2j+1)(2j+3)}}\;A^2 \quad (K=2,\ j_B=j_A=j\ge1), \]
\[ \beta = \frac{A^1}{\sqrt{2}\,\langle B||\boldsymbol\sigma||A\rangle} \quad (K=1), \]
with \( \langle B||\boldsymbol\sigma||A\rangle = 2S_{\kappa\kappa'} \) for a single valence electron (radial overlap dropped by convention); for CI states see CI::sigma_rme().
With \( t = d \) and \( s = h_W \) (PNC, \( k_s = 0 \), so \( K = 1 \) and \( \langle 1\,0\,0\,0|1\,0\rangle = 1 \)), at \( m_A = m_B = m \):
\[ E_{\rm PNC} = A^1_0 = (-1)^{j_B-m} \begin{pmatrix} j_B & 1 & j_A \\ -m & 0 & m \end{pmatrix} A^1 . \]
Examples – static polarisability of Cs 6s, and the 6s-7s PNC amplitude:
The calculations are performed by the Amplitudes library: see Amplitudes::sos_valence(), Amplitudes::ms_valence(), Amplitudes::sos_core(), and Amplitudes::ms_core() for the formulas and conventions (the module, Module::secondOrder, only parses input and prints).
Available options (from ./ampsci -m secondOrder):
1.9.8