Determining the intrinsic reaction coordinate of a chemical reaction by nested path integrals
A computer system can perform a computational methods of determining intrinsic reaction coordinates (IRCs) for chemical transformations (e.g., chemical reactions) via dual nested integration.
1 . A method of determining an intrinsic reaction coordinate (IRC) path from a reactant molecular geometry to a product molecular geometry of a chemical reaction on a potential energy surface, the method comprising:
accessing, by a computer system, data indicating a potential energy surface function and molecular coordinates;
performing, by the computer system, a dual nested integration of two polynomial splines, wherein performing the dual nested integration comprises:
parameterizing a first polynomial spline according to a set of control points corresponding to locations on the potential energy surface of the chemical reaction, the first polynomial spline describing molecular coordinates as a function of a parameter coordinate describing progress along a candidate IRC path from the reactant molecular geometry to the product molecular geometry;
computing the integral of the first polynomial spline, the integrated first polynomial spline describing length along the candidate IRC path as a function of the parameter coordinate;
generating a second polynomial spline by inverting the length and parameter coordinates of the integrated first polynomial spline;
parameterizing the second polynomial spline according to interpolated sampling points along the integral of the first polynomial spline, the second polynomial spline describing the gradient of the potential energy surface of the chemical reaction along the candidate IRC path from the reactant molecular geometry to the product molecular geometry; and
computing an action integral, the action integral being the integral of the second polynomial spline, wherein the integral of the first polynomial spline is the integrand of the second polynomial spline integral;
determining, by the computer system, the IRC path from the reactant molecular geometry to the product molecular geometry based on the action integral, the IRC path indicating a feasibility of the chemical reaction; and
based on the IRC path indicating the chemical reaction is feasible, selecting to perform the chemical reaction to generate the product molecular geometry from the reactant molecular geometry.
2 . The method of claim 1 , wherein determining the IRC path from the reactant molecular geometry to the product molecular geometry of a chemical reaction on a potential energy surface does not include determining the Hessian of an objective function at or between any control point or sampling point, wherein the objective function is the action integral.
3 . The method of claim 1 , wherein the parameter coordinate describing progress along the candidate IRC path is an arbitrary time coordinate t, wherein the first polynomial spline is constrained at t=0 and t=1.
4 . The method of claim 1 , wherein the integral of the first polynomial spline is:
∫
0
t
dt
′
∑
A
(
dz
A
dt
′
)
2
,
wherein z A represents continuous molecular coordinates.
5 . The method of claim 1 , wherein the integral of the second polynomial spline is:
∫
0
l
dl
′
∑
A
(
g
A
(
l
′
)
)
2
,
wherein g A is the second polynomial spline approximating the gradient of the potential energy surface with respect to the molecular coordinates as a function of the length coordinate l′.
6 . The method of claim 1 , wherein the integral of the second polynomial spline is:
∫
0
l
dl
′
∑
B
(
g
B
(
z
B
(
l
′
)
)
)
2
,
wherein l′ is the length coordinate, z B represents continuous molecular coordinates, and g B is the gradient of the potential energy surface.
7 . The method of claim 1 , wherein the method is applied iteratively to candidate IRCs defined by more than one transition state.
8 . The method of claim 1 , wherein the set of control points are uniformly spaced along the first polynomial spline.
9 . The method of claim 1 , wherein determining the IRC path based on the action integral comprises determining the gradient of the action integral along the IRC path.
10 . The method of claim 9 , wherein determining the IRC path based on the action integral further comprises determining the IRC path by reducing the value of the action integral.
11 . The method of claim 10 , wherein the value of the action integral is reduced using gradient-based optimization.
12 . A non-transitory computer-readable storage medium storing instructions for determining an intrinsic reaction coordinate (IRC) path from a reactant molecular geometry to a product molecular geometry of a chemical reaction on a potential energy surface, wherein the instructions, when executed by a computer, cause the computer to perform operations comprising:
accessing, by the computer, data indicating a potential energy surface function and molecular coordinates;
performing, by the computer, a dual nested integration of two polynomial splines, wherein performing the dual nested integration comprises:
parameterizing a first polynomial spline according to a set of control points corresponding to locations on the potential energy surface of the chemical reaction, the first polynomial spline describing molecular coordinates as a function of a parameter coordinate describing progress along a candidate IRC path from the reactant molecular geometry to the product molecular geometry;
computing the integral of the first polynomial spline, the integrated first polynomial spline describing length along the candidate IRC path as a function of the parameter coordinate;
generating a second polynomial spline by inverting the length and parameter coordinates of the integrated first polynomial spline;
parameterizing the second polynomial spline according to interpolated sampling points along the integral of the first polynomial spline, the second polynomial spline describing the gradient of the potential energy surface of the chemical reaction along the candidate IRC path from the reactant molecular geometry to the product molecular geometry; and
computing an action integral, the action integral being the integral of the second polynomial spline, wherein the integral of the first polynomial spline is the integrand of the second polynomial spline integral;
determining, by the computer, the IRC path from the reactant molecular geometry to the product molecular geometry based on the action integral, the IRC path indicating a feasibility of the chemical reaction; and
based on the IRC path indicating the chemical reaction is feasible, selecting to perform the chemical reaction to generate the product molecular geometry from the reactant molecular geometry.
13 . The computer-readable storage medium of claim 12 , wherein determining the IRC path from the reactant molecular geometry to the product molecular geometry of a chemical reaction on a potential energy surface does not include determining the Hessian of an objective function at or between any control point or sampling point, wherein the objective function is the action integral.
14 . The computer-readable storage medium of claim 12 , wherein the parameter coordinate describing progress along the candidate IRC path is an arbitrary time coordinate t, wherein the first polynomial spline is constrained at t=0 and t=1.
15 . The computer-readable storage medium of claim 12 , wherein the integral of the first polynomial spline is:
∫
0
t
dt
′
∑
A
(
dz
A
dt
′
)
2
,
wherein z A represents continuous molecular coordinates.
16 . The computer-readable storage medium of claim 12 , wherein the integral of the second polynomial spline is:
∫
0
l
dt
′
∑
A
(
g
A
(
l
′
)
)
2
,
wherein g A is the second polynomial spline approximating the gradient of the potential energy surface with respect to the molecular coordinates as a function of the length coordinate l′.
17 . The computer-readable storage medium of claim 12 , wherein the integral of the second polynomial spline is:
∫
0
l
dl
′
∑
B
(
g
B
(
z
B
(
l
′
)
)
)
2
,
wherein l is the length coordinate, z B represents continuous molecular coordinates, and g B is the gradient of the potential energy surface.
18 . The computer-readable storage medium of claim 12 , wherein the operations are applied iteratively to candidate IRCs defined by more than one transition state.
19 . The computer-readable storage medium of claim 12 , wherein the set of control points are uniformly spaced along the first polynomial spline.
20 . The computer-readable storage medium of claim 19 , wherein determining the IRC path based on the action integral comprises determining the gradient of the action integral along the IRC path.