Multidimensional modeling of the number of local minimum-energy structures and the energy of putative global minima

Using data about the number of local minimal energy structures and the energy of the global minima structures of the Lennard-Jones potential for 9 ≤ N ≤ 16 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>9</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> atoms and dimension 1 ≤ D ≤ N - 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we propose new expressions to extrapolate these quantities taking advantage of the new multidimensional results. These expressions are obtained based on both multidimensional data fitting and neural networks to dimensionally move from known values, providing then new and more reliable extrapolations of the number of local minima energy structures, compared with previous methods based only on extrapolation of the number of three-dimensional local minimal energy structures for N ≤ 15 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>15</mml:mn> </mml:mrow> </mml:math> atoms. Our estimations of the number of local minima in terms of N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> and D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> show that the maximum number of stable structures of the potential is found at D=3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . We also propose an expression to estimate the energies of the global minima energy structures, beyond the dimension D=3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> case. Surprisingly, we found that for each value of N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> , the energy of the global minimum can be divided into two regimes: when D ≤ \lceil N/2 \rceil <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mo stretchy="false" form="prefix">⌈</mml:mo> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌉</mml:mo> </mml:mrow> </mml:math> and when D ≥ \lceil N/2 \rceil <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≥</mml:mo> <mml:mo stretchy="false" form="prefix">⌈</mml:mo> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌉</mml:mo> </mml:mrow> </mml:math> , where \lceil X\rceil <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">⌈</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false" form="postfix">⌉</mml:mo> </mml:mrow> </mml:math> means the whole part of X <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>X</mml:mi> </mml:math> , each exhibiting a significantly different trend from the other. While the data are limited, the observed trends appear robust, and the resulting fits provide reliable estimates even for values of N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> beyond current computational feasibility.