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general [2019/10/04 16:59] – [Format of Quantum Numbers] admingeneral [2023/11/07 12:40] (current) mueller
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-====== Documentaion for SPFIT and SPCAT ======+====== Documentation for SPFIT and SPCAT ====== 
  
 Last local (HSPM) modification: May 2, 2001\\ Last local (HSPM) modification: May 2, 2001\\
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 Some of the details of the program are described in Some of the details of the program are described in
-  H. M. Pickett, "The Fitting and Prediction of Vibration-Rotation Spectra with Spin Interactions," J. Mol. Spectros. 148, 371-377 (1991).+<alert type="info"> 
 +H. M. Pickett, "The Fitting and Prediction of Vibration-Rotation Spectra with Spin Interactions," J. Mol. Spectrosc. 148, 371-377 (1991). 
 +</alert>
  
 ===== Format of Quantum Numbers ===== ===== Format of Quantum Numbers =====
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 The length of the quantum number list is determined by the number of spins requested. The length of the quantum number list is determined by the number of spins requested.
 The factoring of the Hamiltonian is determined by the parameter set. The factoring of the Hamiltonian is determined by the parameter set.
- 
- 
  
 ===== Format of the ''lin'' File ===== ===== Format of the ''lin'' File =====
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 The freeform input begins in column 37 and extends to the end of the line. The freeform input begins in column 37 and extends to the end of the line.
 See the notes at the end of the next section for more on the freeform input. See the notes at the end of the next section for more on the freeform input.
 +
 ===== Format of the ''par'' and ''var'' Files ===== ===== Format of the ''par'' and ''var'' Files =====
  
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 **Option information beginning on line 3:** CHR, SPIND, NVIB, KNMIN, KNMAX, IXX, IAX, WTPL, WTMN, VSYM, EWT, DIAG **Option information beginning on line 3:** CHR, SPIND, NVIB, KNMIN, KNMAX, IXX, IAX, WTPL, WTMN, VSYM, EWT, DIAG
  
-> CHR: character to modify parameter names file (must be in first column) sping.nam , default is **g**. **a** is used for Watson A set, **s** is used for Watson S set. Other character replaces the **g** in the name 'sping'. Only used to label the .fit output file. (Ignored on all but first option line.) SPFIT looks for the //nam// files in the current directory and then in the path given by the SPECNAME environment variable. (i.e. put something like SET SPECNAME=C:\SPECTRA\ in AUTOEXEC.BAT for Windows or setenv SPECNAME /spectra/ for unix). The trailing path delimiter is required. +> CHR: character to modify parameter names file (must be in first column) sping.nam , default is **g**. **a** is used for Watson A set, **s** is used for Watson S set. Other character replaces the **g** in the name 'sping'. Only used to label the .fit output file. (Ignored on all but first option line.) SPFIT looks for the //nam// files in the current directory and then in the path given by the SPECNAME environment variable. (i.e. put something like SET SPECNAME=C:\SPECTRA\ in AUTOEXEC.BAT for Windows or setenv SPECNAME /spectra/ for unix). The trailing path delimiter is required.\\ 
- +> sign SPIND: If negative, use symmetric rotor quanta. If positive, use asymmetric rotor quanta (Sign ignored on all but first option line.)\\ 
-> sign SPIND: If negative, use symmetric rotor quanta. If positive, use asymmetric rotor quanta (Sign ignored on all but first option line.) +> mag SPIND = degeneracy of spins, first spin degeneracy in units digit, second in tens digit, etc. (If last digit is zero, spin degeneracies occupy two decimal digits and the zero is ignored.)\\ 
- +> sign NVIB: positive means //I<sup>r</sup>// representation (z = a, x = b, y = c), usually used for prolate rotors; negative means //III<sup>l</sup>// representation (z = c, x = b, y = a), usually used for oblate rotors. (Sign ignored on all but first option line.)\\ 
-> mag SPIND = degeneracy of spins, first spin degeneracy in units digit, second in tens digit, etc. (If last digit is zero, spin degeneracies occupy two decimal digits and the zero is ignored.) +> mag NVIB = number of states (e. g. vibronic; also possible: isotopomers etc.; **counted from zero !**) on the first option line, identity of the vibronic state on all but the first option line. (max. value = 99)\\ 
- +> KNMIN,KNMAX = minimum and maximum K values. If both = 0, then linear molecule is selected.\\ 
-> sign NVIB: positive means //I<sup>r</sup>// representation (z = a, x = b, y = c), usually used for prolate rotors; negative means //III<sup>l</sup>// representation (z = c, x = b, y = a), usually used for oblate rotors. (Sign ignored on all but first option line.) +> IXX: binary flag for inclusion of interactions: 1 means no delta N, 2 means no delta J, 4 means no delta F<sub>1</sub> ,etc. [default = 0 includes all interactions] (Ignored on all but first option line.)\\ 
- +> sign IAX: If negative, use I<sub>tot</sub> basis in which the last two spins are summed to give > > Itot, which is then combined with the other spins to give F (Sign ignored on all but first option line.)\\
-> mag NVIB = number of states (e. g. vibronic; also possible: isotopomers etc.; **counted from zero !**) on the first option line, identity of the vibronic state on all but the first option line. (max. value = 99) +
- +
-> KNMIN,KNMAX = minimum and maximum K values. If both = 0, then linear molecule is selected. +
- +
-> IXX: binary flag for inclusion of interactions: 1 means no delta N, 2 means no delta J, 4 means no delta F<sub>1</sub> ,etc. [default = 0 includes all interactions] (Ignored on all but first option line.) +
- +
-> sign IAX: If negative, use I<sub>tot</sub> basis in which the last two spins are summed to give > > Itot, which is then combined with the other spins to give F (Sign ignored on all but first option line.) +
 > WTPL,WTMN = statistical weights for even and odd state > WTPL,WTMN = statistical weights for even and odd state
-> mag IAX = axis for statistical weight ( 1=a, 2=b, 3=c, add 3 if K-odd are excluded, add 6 if K-even are excluded) +> mag IAX = axis for statistical weight ( 1=a, 2=b, 3=c, add 3 if K-odd are excluded, add 6 if K-even are excluded)\\ 
- +> VSYM: If positive, vibronic symmetry coded as decimal digits (odd digit means reverse WTPL with WTMN) example: 10 = ( v=0 even, v=1 odd) (Only works for the first nine states) (Value ignored on all but first option line.) If negative, signal that the next line is also an option line.\\
-> VSYM: If positive, vibronic symmetry coded as decimal digits (odd digit means reverse WTPL with WTMN) example: 10 = ( v=0 even, v=1 odd) (Only works for the first nine states) (Value ignored on all but first option line.) If negative, signal that the next line is also an option line. +
 > EWT = EWT0 + EWT1*100 = weight for states with 3-fold E symmetry. Ignore if EWT is negative (default) (WTPL and WTMN apply to A1 and A2 symmetry) > EWT = EWT0 + EWT1*100 = weight for states with 3-fold E symmetry. Ignore if EWT is negative (default) (WTPL and WTMN apply to A1 and A2 symmetry)
    
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 > EWT0 = (2I+1)(I+1)(4I)/3 > EWT0 = (2I+1)(I+1)(4I)/3
  
-**NOTE:** These weights can be divided by a common multiple if the rotational partition function is divided by the same factor. The A1 and A2 states are for MOD(ABS(K)–EWT1,3) = 0 with EWT1 = 0 for l = 0, EWT1 = 1 for l = 1, and EWT2 = 2 for l = –1. STATES with EWT1 not zero MUST be specified in adjacent pairs. E symmetry states will be designated with positive K for symmetric top quanta. For asymmetric top quanta with l = 1, K<sub>a</sub> + K<sub>c</sub> = N+1. For asymmetric top quanta with l = –1, K<sub>a</sub> + K<sub>c</sub> = N. This designation for quanta in l=1 and l = –1 states will also be applied to A symmetry states if there are only delta l = 0 operators. If both WTPL and WTMN are not zero, there will be two E states with the same nominal quantum number. (CALMRG will merge the degenerate transitions into a single line. (It appears as if this does not always happen).)\\ +<alert type="warning"> 
- +**NOTE:** These weights can be divided by a common multiple if the rotational partition function is divided by the same factor. The A1 and A2 states are for MOD(ABS(K)–EWT1,3) = 0 with EWT1 = 0 for l = 0, EWT1 = 1 for l = 1, and EWT2 = 2 for l = –1. STATES with EWT1 not zero MUST be specified in adjacent pairs. E symmetry states will be designated with positive K for symmetric top quanta. For asymmetric top quanta with l = 1, K<sub>a</sub> + K<sub>c</sub> = N+1. For asymmetric top quanta with l = –1, K<sub>a</sub> + K<sub>c</sub> = N. This designation for quanta in l=1 and l = –1 states will also be applied to A symmetry states if there are only delta l = 0 operators. If both WTPL and WTMN are not zero, there will be two E states with the same nominal quantum number. (CALMRG will merge the degenerate transitions into a single line. **(It appears as if this does not always happen)**.) 
 +</alert>
 <code> <code>
 DIAG = DIAG =
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 **NOTE:** For many cases only a single option line is needed. If different vibronic states have different spin multiplicity or different KMIN, KMAX additional lines are needed. Note that additional lines are signaled by the sign of VSYM. The first option line sets up the defaults for all the vibrational states, and subsequent option lines specify deviations from the default. It is possible to mix Boson and Fermion states in the same calculation, e.g. fitting different isotopomers together, but the quantum number format (QNFMT) in SPCAT output will be correct only for the v = 0 state. **NOTE:** For many cases only a single option line is needed. If different vibronic states have different spin multiplicity or different KMIN, KMAX additional lines are needed. Note that additional lines are signaled by the sign of VSYM. The first option line sets up the defaults for all the vibrational states, and subsequent option lines specify deviations from the default. It is possible to mix Boson and Fermion states in the same calculation, e.g. fitting different isotopomers together, but the quantum number format (QNFMT) in SPCAT output will be correct only for the v = 0 state.
 </alert> </alert>
-==== Parameter lines: IDPAR, PAR, ERRPAR / LABEL ====+ 
 +====== Coding of the Parameters ====== 
 + 
 +===== Parameter lines: IDPAR, PAR, ERRPAR / LABEL =====
  
 where IDPAR is a parameter identifier, PAR is the parameter value,  ERRPAR is the parameter uncertainty, LABEL is a parameter label (10 characters are used) that is delimited by **/. ** If the sign of IDPAR is negative, SPFIT constrains the ratio of this parameter to the previous parameter to a fixed value during the fit. where IDPAR is a parameter identifier, PAR is the parameter value,  ERRPAR is the parameter uncertainty, LABEL is a parameter label (10 characters are used) that is delimited by **/. ** If the sign of IDPAR is negative, SPFIT constrains the ratio of this parameter to the previous parameter to a fixed value during the fit.
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 PARAMETER identifiers (IDPAR) are coded in decimal digitform in the order PARAMETER identifiers (IDPAR) are coded in decimal digitform in the order
 <code> <code>
-NFF, I2, I1, NS, TYP, KSQ, NSQ, V2, V1\\ +NFF, I2, I1, NS, TYP, KSQ, NSQ, V2, V1 
-for NVIB < 10 each element occupies one digit except TYP which occupies two digits, i.e.\\ + 
-(((((((FF*10+I2)*10+I1)*10+NS)*100+TYP)*10+KSQ)*10+NSQ)*10+V2)*10+V1\\ +for NVIB < 10 each element occupies one digit except TYP which occupies two digits, i.e. 
-for NVIB > 9: each element occupies one digit except TYP, V1, and V2 which occupy two digits, i.e.\\ +    (((((((FF*10+I2)*10+I1)*10+NS)*100+TYP)*10+KSQ)*10+NSQ)*10+V2)*10+V1 
-(((((((FF*10+I2)*10+I1)*10+NS)*100+TYP)*10+KSQ)*10+NSQ)*100+V2)*100+V1+ 
 +for NVIB > 9: each element occupies one digit except TYP, V1, and V2 which occupy two digits, i.e. 
 +    (((((((FF*10+I2)*10+I1)*10+NS)*100+TYP)*10+KSQ)*10+NSQ)*100+V2)*100+V1
 </code> </code>
   * NFF = Fourier flag (used for internal rotation) If NFF < 11, basic operator is multiplied by cos (NFF * 2p K<sub>avg</sub>r / 3) else operator multiplied by sin ( (NFF-10) * 2p K<sub>avg</sub>r / 3) where r is coded by the absoulte value of parameter ID=9100vv. See further discussion below.   * NFF = Fourier flag (used for internal rotation) If NFF < 11, basic operator is multiplied by cos (NFF * 2p K<sub>avg</sub>r / 3) else operator multiplied by sin ( (NFF-10) * 2p K<sub>avg</sub>r / 3) where r is coded by the absoulte value of parameter ID=9100vv. See further discussion below.
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 If IDPAR is less than zero the magnitude is taken. In SPFIT, the parameter value will be constrained to be a constant ratio of the preceding parameter value. In this way linear combinations of parameters can be fit as a unit. If IDPAR is less than zero the magnitude is taken. In SPFIT, the parameter value will be constrained to be a constant ratio of the preceding parameter value. In this way linear combinations of parameters can be fit as a unit.
- 
-===== Coding of the Parameters ===== 
  
 ===== Format of the ''int'' File ===== ===== Format of the ''int'' File =====
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 **NOTE:** Dipoles with SYM > 0 are assumed to be in units of Debye. Dipoles with SYM = 0 are assumed to be in units of a Bohr magneton. Dipoles which are even order in direction cosine or N are assumed to be imaginary, except between states with EWT1 = 1. Dipoles between states with EWT1 = (0,2), (2,0), and (2,2) are ignored, but the matrix elements are calculated using corresponding dipoles from states with EWT1 = 1 (see below). For ITYP = 7 or ITYP = 8, I1 is used for the Fourier order and not the spin type. The constant r is specified in the parameter set. The sign of the r parameter is used to designate a special symmetry for the Fourier series.  If this sign is different for V1 and V2, then 0.5 is subtracted from the Fourier order.  For example, if IDIP = 72012, the basic //b//-dipole operator is multiplied by cos ( 3p K<sub>avg</sub>r / 3) instead of cos (4p K<sub>avg</sub>r / 3).  If the magnitude of  r is not the same for the two states, replace K<sub>avg</sub>r with (K<sub>1</sub>r<sub>1</sub> + K<sub>2</sub>r<sub>2</sub>) / 2. ITYP = 8 (with I1 > 0) dipoles are multiplied by //i//, and the symmetry of the states connected is 3 – SYM and the units follow the state symmetry (e.g. 81000 is in Debye ). ITYP = 2, 5 are used for first-order Herman-Wallis corrections. ITYP = 3, 4, 6, 11, 12 are used for second-order Herman-Wallis corrections. **NOTE:** Dipoles with SYM > 0 are assumed to be in units of Debye. Dipoles with SYM = 0 are assumed to be in units of a Bohr magneton. Dipoles which are even order in direction cosine or N are assumed to be imaginary, except between states with EWT1 = 1. Dipoles between states with EWT1 = (0,2), (2,0), and (2,2) are ignored, but the matrix elements are calculated using corresponding dipoles from states with EWT1 = 1 (see below). For ITYP = 7 or ITYP = 8, I1 is used for the Fourier order and not the spin type. The constant r is specified in the parameter set. The sign of the r parameter is used to designate a special symmetry for the Fourier series.  If this sign is different for V1 and V2, then 0.5 is subtracted from the Fourier order.  For example, if IDIP = 72012, the basic //b//-dipole operator is multiplied by cos ( 3p K<sub>avg</sub>r / 3) instead of cos (4p K<sub>avg</sub>r / 3).  If the magnitude of  r is not the same for the two states, replace K<sub>avg</sub>r with (K<sub>1</sub>r<sub>1</sub> + K<sub>2</sub>r<sub>2</sub>) / 2. ITYP = 8 (with I1 > 0) dipoles are multiplied by //i//, and the symmetry of the states connected is 3 – SYM and the units follow the state symmetry (e.g. 81000 is in Debye ). ITYP = 2, 5 are used for first-order Herman-Wallis corrections. ITYP = 3, 4, 6, 11, 12 are used for second-order Herman-Wallis corrections.
 </alert> </alert>
 +
 ===== Format of the ''cat'' File ===== ===== Format of the ''cat'' File =====
  
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 > QN(6) = Quantum numbers for the state > QN(6) = Quantum numbers for the state
  
-===== Special  Considerations for Linear Molecules =====+====== Special  Considerations for Linear Molecules =====
 + 
 +This program set will calculate a variety of interactions and transitions within a Hund's case(b) basis, including spin orbit interactions which change spin multiplicity.  The operator N<sub>a</sub> in the asymmetric rotor becomes lambda for the linear molecule. N is the sum of rotational and electronic orbital angular momenta. For linear molecules, it is convenient (but not essential) to think of the angular momentum along the bond as being purely electronic in nature. In the asymmetric rotor language of this program, the first-order spin orbit interaction takes the operator form of a vector dot product of a direction cosine with the spin vector.  It can have two distinct symmetries: S·f<sub>a</sub> connects states of the same lambda, while S·f<sub>b</sub> and S·f<sub>c</sub> conect states where lambda differs by one. For S·f<sub>b</sub> or S·f<sub>c</sub> , the L<sub>b</sub> or L<sub>c</sub> operator is implicitly included in the parameter. When the spin orbit operator connects different spin multiplicity, the reduced matrix value of <S||**S**||S'> is set to unity. 
 + 
 +Use of the symmetries in this program takes some care, particularly for linear molecules where it may not be immediately obvious whether to use the b or the c axis to designate perpendicular operators. For consistency with the parity designation for the symmetric top quanta, the vibronic wave function should be chosen so that it is symmetric with respect to the //ab// plane. Then the //b// axis can be used for the inversion defining axis (i.e. IAX = 2 in the option lines of the .par and .var files can be used to define selection rules under inversion).  With this choice, the symmetry of rotation lines in the D<sub>2</sub> group are: 
 + 
 +|A   |even N for S and all N for all other even l (even parity l doublet)| 
 +|B(a)|odd N for S and all N for all other even l (odd parity l doublet) 
 +|B(b)|all N for all odd l (even parity l doublet)                        | 
 +|B(c)|all N for all odd l (odd parity l doublet)                         | 
 + 
 +For S<sup>+</sup> states, the parity is odd for odd N, while for S<sup>-</sup> states the parity is odd for even N. This means that the Hamiltonian can couple S<sup>+</sup> with S<sup>–</sup> via an operator of  B(a) symmetry or B(c) symmetry. An example is an operator like 10200001, which is the S·f<sub>a</sub> spin orbit interaction operator between state v = 0 and v = 1.  For normal coupling not involving S<sup>-</sup> states (or for coupling between S<sup>-</sup> states) the operators should have A or B(b) symmetry.  Similarly, electric dipole transitions with  Dl even should have B(a) symmetry, and transitions with  Dl odd should have B(c) symmetry so that the parity changes sign with the transition. Magnetic dipole transitions should follow the Hamiltonian symmetry. 
 + 
 +The g, u symmetry for an electronic state is for the parity of the wave-function under inversion of the space fixed axes. The nuclear exchange symmetry, on the other hand, affects only the statistical weights and does not have any further impact on the factoring of the Hamiltonian. In general, if IAX = 2, WTPL will be the nuclear spin weight for the A and  B(b) states, while WTMN will be the weight for the other two symmetries. For S<sup>+</sup><sub>g</sub> S<sup>-</sup><sub>u</sub> and g states with other l, WTPL is the weight for even permutations, while for S<sup>-</sup><sub>g</sub> S<sup>+</sup><sub>u</sub> and u states with other l, WTPL is the weight for odd permutations. For example, in oxygen, S<sup>+</sup><sub>g</sub> and D<sub>g</sub> have WTPL = 1 and WTMN = 0, while S<sup>-</sup><sub>g</sub> and D<sub>u</sub> have WTPL = 0 and WTNM = 1. The dipole types given above provide for both allowed and forbidden transitions. For transitions that owe their intensity to spin orbit interactions, the effective transition moment with be the product of the interaction and a transition moment to some intermediate state. Examples are a S<sup>-</sup><sub>g</sub>D electric dipole code of 21vv'0 and a magnetic dipole code of 11vv'1. Electric dipole transitions between S<sup>+</sup> and S<sup>-</sup> will use 11vv'0, while magnetic transitions will use 1vv'1. For transitions between S<sup>+</sup> and S<sup>+</sup> the roles of these operators are reversed. Note that for S S transitions, 11vv'0 has selection rules of DN = –2, 0, +2, while 1vv'1 has selection rules of DN = –1, +1. 
 + 
 +The correlation between parity and e,f designations follow the recommendations of J. M. Brown //et al.//, //J. Mol. Spectrosc.// **55,** 500 (1975). 
 + 
 +| |odd spin multiplicity   |even spin multiplicity| 
 +|e|p = (–1)<sup>J–1/2</sup>|p = (–1)<sup>J+1</sup>
 +|f|p = (–1)<sup>J+1/2</sup>|p = (–1)<sup>J </sup>
 + 
 +An example of .var file for oxygen like molecule: 
 + 
 +''%%mock oxygen states%%''\\ 
 +''%%4%%''\\ 
 +''%%-3 3 0 0 0 2 0 1 -1 /default for v=0 triplet Sigma-(g)%%''\\ 
 +''%% 1 1 2 2 0 2 1 0 -1 /v=1 is singlet Delta(g)%%''\\ 
 +''%% 1 2 0 0 0 2 1 0  0 /v=2 is singlet Sigma-(g)%%''\\ 
 +''%%       11  1e+7   0 /term value for v=1%%''\\ 
 +''%%       22  2e+7   0 /term value for v=2%%''\\ 
 +''%%110010000  1000.0 0 /spin-spin interaction for v=0%%''\\ 
 +''%%      199 10000.0 0 /B for all v%%'' 
 + 
 +An example of .int file for oxygen like molecule: 
 + 
 +''%%mock oxygen rotational and electronic transitions%%''\\ 
 +''%%101 99000 200. 0 6 -80. -80. 99999999.%%''\\ 
 +''%% 1000 1. /magnetic moment v=0%%''\\ 
 +''%% 1110 1. /magnetic moment v=1%%''\\ 
 +''%%11011 1. /sigma - delta magnetic moment%%''\\ 
 +''%% 1021 1. /sigma - sigma magnetic moment%%'' 
 + 
 +The quantum number correlations between Hund's case (b) and case (a) can be a bit confusing at first. For example in a doublet P state N=J-1/2 always correlates with W =3/2 and N=J+1/2 always correlates with W =1/2 on the basis of projection. For A < 0, e.g. OH, the projection-based correlation follows the energy ordering.  For A>0, the lower energy state is N=J+1/2 and W =3/2 as long as J+1/2 < sqrt(A/2B). Above this J,  N = J+1/2 and W =1/2 (based on projection) is higher in energy than and N=J-1/2 and W = 3/2.  Therefore quantum number assignments based on projections lead to different quanta than those based on energy.  For a triplet S state, N=J+1 correlates with S =0 based on projection, N=J correlates with an odd combination of S =1 and S =-1, and N=J-1 correlates with an even combination of S =1 and S =-1. 
 + 
 +Since q multiplies the same operator as (B-C) / 2, it is possible to use the sign of q to determine whether there are more electrons in the ab plane (q > 0) or whether there are more electrons in the ac plane (q < 0). 
 + 
 +Explicit approximate relationships for the parameters are: 
 + 
 +|100100vv' |A                  | 
 +|100101vv' |2 A<sub>J</sub>    | 
 +|1vv'      |B                  | 
 +|100400vv' |–p/              | 
 +|400vv'    |q/2                | 
 +|200100vv' |a                  | 
 +|1200100vv'|c                  | 
 +|1200000vv'|b + c/3            | 
 +|1200400vv'|–d/              | 
 +|2200100vv'|1.5eQq<sub>1</sub>
 +|2200400vv'|–eQq<sub>2</sub>/4 | 
 +|1100100vv'|4 l S (2S–1)       | 
 + 
 +The extra factors of S in the definition of the spin-spin interaction parameter l,  i. e. a spin-spin interaction  2 l ( S<sub>z</sub><sup>2</sup> – S <sup>2</sup> /  3), is a correction for the special normalization assumed for eQq. 
 + 
 +====== Special  Considerations for 'l'-doubled States ====== 
 + 
 +The  //l//-doubled states must be specified in adjacent pairs. The EWT1 = 1 states are those with //K l// > 0, and EWT1 = 2 states are those with //K  l// <= 0. The sign of //K// represents the parity, as in the non //l//-doubled states. Operators should be only specified between vibrational states with EWT1 = (0,0), (0,1), (1,0), (1,1), (1,2), and (2,1). Operators between vibrational states with EWT1 = (0,2), (2,0), and (2,2) are ignored. Operators connecting vibrational states with different //'l'// obey the selection rule that '//K-l//' can only change by multiples of 3. Operators diagonal in //l// have no '//K-l//' selection rules. If EWT1 = 1 for both states and if the parameter would normally be implicitly imaginary (i.e. operators odd-order in angular momentum for the Hamiltonian, or even-order for the dipole moment), then the parameter is assumed to be real and the rotational operator is multiplied by the sign of '//l//<sub>z</sub>'. DIAG = 0 is not recommended on the first option line in the //par// file, since the first-order energy is not likely to ordered with //K.// 
 + 
 +The //K// quantum numbers for  //l//-doubled states are designated specially when asymmetric rotor quanta are used so that the lower //K// doublet is associated with the EWT1 = 1 state and the upper //K// doublet and //K// = 0 states are assiciated with EWT1 = 2.  In this way the degenerate states have the same quantum numbers. 
 + 
 +=====  Simple Examples ===== 
 + 
 +Example of parameter types for asymmetric rotors (assuming < 10 vibronic states): 
 + 
 +|11       |energy for v = 1                                                     | 
 +|01       |first order Fermi (F<sub>0</sub>) interaction between v = 0 and v = 1| 
 +|10000    |A<sub>00</sub>                                                       | 
 +|10099    |A (for all vibrational states)                                       | 
 +|20099    |B (dito)                                                             | 
 +|30099    |C (dito)                                                             | 
 +|40099    |0.25*(B – C)  (if prolate basis selected)                            | 
 +|299      |–D<sub>J</sub>                                                       | 
 +|1199     |–D<sub>JK</sub>                                                      | 
 +|2000     |–D<sub>K</sub> for v = 0                                             | 
 +|600001   |i N<sub>c</sub> interaction between v = 0 and v = 1                  | 
 +|20000099 |N·I for second spin                                                  | 
 +|120010099|S<sub>a</sub> I<sub>a</sub>                                          | 
 +|220010099|1.5*c<sub>zz</sub> for second spin                                   | 
 +|220040099|0.25*(c<sub>xx</sub>–c<sub>yy</sub>) for second spin                 | 
 + 
 +Quadrupole and magnetic spin-spin interactions are defined to be traceless (i.e. c<sub>xx</sub> + c<sub>yy</sub> + c<sub>zz</sub> = 0 or T<sub>xx</sub> + T<sub>yy</sub> + T<sub>zz</sub> = 0). Therefore, all three components cannot be fit simultaneously. The most efficient choice of parameters is shown in the table below. In cases where the user wants an alternative, it is possible to use constrained parameters. For example, to fit c<sub>aa</sub> and c<sub>cc</sub> (with no multipliers):
  
-===== Special  Considerations for 'l'-doubled States =====+''%% 220010099   100.%%''\\ 
 +''%%–220020099  –100.%%''\\ 
 +''%% 220030099    50.%%''\\ 
 +''%%–220020099   –50.%%''
  
-===== Some Examples =====+specifies c<sub>aa</sub> 100, c<sub>cc</sub> 50, and c<sub>bb</sub> –150.
  
-===== Installation Instructions =====+====== Installation Instructions ======
  
 The Makefile shows how the various files are to be linked. The programs have been tested with Microsoft Visual The Makefile shows how the various files are to be linked. The programs have been tested with Microsoft Visual
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