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A perfect in phase FD algorithm for problems in quantum chemistry

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Our research pays attention to the deployment of newly algorithm which is useful on quantum chemical problems.

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$$\begin{aligned} { LTE}_{CL}= & {} { LTE}_{NM122S3SPD1} = { LTE}_{NM122S3SPD2} = { LTE}_{NM122S3SPD3} \\= & {} { LTE}_{NM122S3SPD4} = { LTE}_{NM122S3SPD5} \approx \\\approx & {} h^{12} \, \varpi _{0} = h^{12} \, \Biggl [ -{\frac{13\,{ iiivxvx} \left( x \right) \tau \left( x \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{136080}}\\&- {\frac{19\, \left( {{ iiivxvx}} \left( x \right) \right) ^{4}\tau \left( x \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{4790016}}\\&- {\frac{13\, \left( {{ iiivxvx}} \left( x \right) \right) ^{3}\tau \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{2}}{1197504}}\\&- {\frac{ \left( {{ iiivxvx}} \left( x \right) \right) ^{4} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) }{798336}}\\&- {\frac{5\,{ iiivxvx} \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{3}}{199584}}\\&- {\frac{31\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) \right) ^{2}}{266112}}\\&- {\frac{ \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) }{16632}}\\&- {\frac{109\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) }{1197504}}\\&- {\frac{ \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{2}\tau \left( x \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) }{19008}}\\&- {\frac{19\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}{} { iiivxvx} \left( x \right) }{443520}}\\&- {\frac{7\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}{} { iiivxvx} \left( x \right) }{342144}}\\&- {\frac{157\, \left( {{ iiivxvx}} \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}{} { iiivxvx} \left( x \right) }{11975040}}\\&- {\frac{{ iiivxvx} \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}{} { iiivxvx} \left( x \right) }{187110}}\\&- {\frac{23\,{ iiivxvx} \left( x \right) \tau \left( x \right) {\frac{{\mathrm{d}}^{8}}{{\mathrm{d}}{x}^{8}}}{} { iiivxvx} \left( x \right) }{11975040}}-{\frac{5\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) \right) ^{3}\tau \left( x \right) }{177408}}\\&- {\frac{ \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{4}\tau \left( x \right) }{85536}}-{\frac{ \left( {\frac{{\mathrm{d}}^{9}}{{\mathrm{d}}{x}^{9}}}{} { iiivxvx} \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) }{2395008}}\\&- {\frac{ \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) \right) ^{2}\tau \left( x \right) }{114048}} - {\frac{31\, \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) \right) \tau \left( x \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}{} { iiivxvx} \left( x \right) }{1995840}}\\&- {\frac{17\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) \right) \tau \left( x \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}{} { iiivxvx} \left( x \right) }{1596672}}\\&- {\frac{13\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) \tau \left( x \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}{} { iiivxvx} \left( x \right) }{2395008}}\\&-{\frac{43\,{ iiivxvx} \left( x \right) \tau \left( x \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) \right) ^{2}}{748440}}\\&- {\frac{743\,{ iiivxvx} \left( x \right) \tau \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) ^{2}{\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{5987520}}\\&- {\frac{ \left( {\frac{{\mathrm{d}}^{10}}{{\mathrm{d}}{x}^{10}}}{} { iiivxvx} \left( x \right) \right) \tau \left( x \right) }{23950080}}\\&- {\frac{323\,{ iiivxvx} \left( x \right) \tau \left( x \right) \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}{} { iiivxvx} \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) }{5987520}}\\&- {\frac{5\, \left( {{ iiivxvx}} \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) }{598752}}\\&- {\frac{ \left( {{ iiivxvx}} \left( x \right) \right) ^{6}\tau \left( x \right) }{23950080}}-{\frac{1201\, \left( {{ iiivxvx}} \left( x \right) \right) ^{2}\tau \left( x \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) \right) ^{2}}{23950080}}\\&- {\frac{239\, \left( {{ iiivxvx}} \left( x \right) \right) ^{2}\tau \left( x \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}{} { iiivxvx} \left( x \right) }{23950080}}\\&- {\frac{37\, \left( {{ iiivxvx}} \left( x \right) \right) ^{3}\tau \left( x \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) }{2993760}}\\&- {\frac{73\,{ iiivxvx} \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{598752}}\\&- {\frac{5\, \left( {{ iiivxvx}} \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{99792}}\\&- {\frac{313\, \left( {{ iiivxvx}} \left( x \right) \right) ^{2}\tau \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) }{3991680}}\\&- {\frac{23\,{ iiivxvx} \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\tau \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{} { iiivxvx} \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) }{299376}}\\&- {\frac{353\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}{} { iiivxvx} \left( x \right) \right) \tau \left( x \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}{} { iiivxvx} \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{} { iiivxvx} \left( x \right) }{2395008}} \Biggr ] \end{aligned}$$

where \(\tau \left( x \right) = \tau _{n}\).

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Qiu, J., Huang, J. & Simos, T.E. A perfect in phase FD algorithm for problems in quantum chemistry. J Math Chem 57, 2019–2048 (2019). https://doi.org/10.1007/s10910-019-01061-w

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