[Title]   Any input line beginning with '#' followed by at least one white-space character is a title line, which is transferred to the head part of bx1HRsol or bx1VIBsol input file 'basefilename.inp'.

[Comment]   Any text from an exclamation mark, '!', to the end of the line is regarded as comments and ignored, except in the 'continue' block, in which any text including '!' is copied to the input for bx1HRsol or bx1VIBsol.

[Blank line]   Any blank line consisting of white-space characters and comment part only is skipped by the input parser.

  An option to control the output of 'basefilename.inp' can be specified as;

outputOption potential-type

where potential-type must be either 'powerSeries' or 'FourierSeries' for the input for bx1VIBsol or bx1HRsol, respectively.   It should be noted that bx1fitPlls can use any function consist of the function listed below (constant, power, sine, cosine, logarithm), bx1VIBsol can only use a power series potential and, for bx1HRsol, it must be a Fourier series.

  A function in the following form can be specified as a regression function. Here, a₁, a₂, ... , a_n are fitting parameters which are optimized in the regression analysis, while c₁, ... , c_m are constants.   f₁(x), f₂(x), ..., f_n(x) and g₁(x), ..., g_m(x) are mathematical functions of x, and each can be one of x^k, cos(kx), sin(kx), ln(x), and constant (= 1).

  An 'optFuncs' block is used to specify the parameter-terms, a_i f_i(x), as;

optFuncs{
  func-1  (order-1)  [mult-1]
  func-2  (order-2)  [mult-2]
  ...
}

where func-i, order-i, and mult-i are function type, order, and multiplication factor, respectively, of the i'th function.   Below is an example input for a second-order polynomial,
y = a₀ + a₁ x + a₂ x²;

optFuncs{
  const
  pow  1
  pow  2
}

The function type, func-i, must be one of 'pow' [x^k], 'cos' [cos(kx)], 'sin' [sin(kx)], 'ln' [ln(x)] (natural logarithm), and 'const' [1] (constant).   The order input, order-i, is necessary for pow, cos, and sin.   The i'th function value is multiplied by mult-i when it is specified.   For example,

optFuncs{
  pow  2  0.5
}

specifies a harmonic oscillator function,   y = (1/2)kx².   Clearly, this is a redundant parameter since the same problem can be also solved by specifying a function,   y = ax², and, afterward, the force constant, k, can be derived from a using   a = (1/2)k.   However, it may be useful to retain the physical meaning of the parameter.

  Optionally, constant-terms, c_j g_j(x) can be specified by 'fixFuncs' block as;

fixFuncs{
  c-1  func-1  (order-1)  [mult-1]
  c-2  func-2  (order-2)  [mult-2]
  ...
}

where c-i is the constant coefficient.   The example below is a second-order polynomial, but the coefficient for the first-order term is fixed to 0.324.

optFuncs{
  const
  pow  2
}
fixFuncs{
  0.324  pow  1
}

  The data points used in the linear least-squares regression should be prepared in a 'xyTable' block as follows.

xyTable{
  [deriv_order-1]  x-1  dny-1  [w-1]
  [deriv_order-2]  x-2  dny-2  [w-2]
  ...
}

where deriv_order-i, x-i, dny-i, and w-i are order of the derivative, x (independent variable), y (dependent variable) or its derivative, and weight, respectively, for the regression analysis.   Below is a simple examples containing the x-y data only.

xyTable{
  -1.02   122.8
   0.03   0.
   1.11   111.2
}

  In many cases for intramolecular motions, the location of a stationary point, x_s, is a useful and important parameter of the potential energy curve.   Such information can be input as a datum like,   dy/dx = 0 at x = x_s.   Also useful results of frequency analysis can be specified as second derivative, d²y/dx², input.   Derivatives can be specified by 'deriv_order-i' optional term, for which 'deriv' or 'deriv1' stands for the first derivative, 'deriv2' for second derivative, etc.   The following is an example using first and second derivative at x = 0.03.

xyTable{
         -1.02   122.8
          0.03   0.
  deriv1  0.03   0.
  deriv2  0.03   229.1
          1.11   111.2
}

When the third field, w-i, is present, it is used to weight the regression analysis. Default weight is unity (1.0).

  The content of a continue block, which begins with 'continue{' and terminates with '}', is nothing to do with the bx1fitPlls, but are merely transferred to the input for a subsequent tool, bx1VIBsol or bx1HRsol.   It is a convenience for repeated fit-and-solve procedures in which one may need to repeat manual 'copy & paste' tasks without this feature.

  Details of the linear least-squares regression analysis are printed to the console (standard output).   Use redirection to store these into a file.   Below is an example output.

===== results of linear least squares regression =====
Function:
 y = a0 + a1 * x + a2 * x^2
 y' = a1 + a2 * 2*x
 y" = a2 * 2
Optimum parameters (+-) standard deviations (relative)
 a0 = -6.7444281e+000 (+-)  5.7137802e+000 (  84.72 %)
 a1 = -1.2905306e+001 (+-)  4.7173646e+000 (  36.55 %)
 a2 =  1.1251074e+002 (+-)  3.8602164e+000 (   3.43 %)
Parameter correlation coefficients
           a0       a1
 a1   0.00229
 a2  -0.50969 -0.06898

  An input file for bx1VIBsol, basefilename.inp, is created when outputOption powerSeries is found in the bx1fitPlls input and the regression function is confirmed to be a power series.   Similarly, outputOption FourierSeries option is used to prepare a Fourier series input for bx1HRsol.   See reference manuals for bx1VIBsol and bx1HRsol for details of these input files.