Analytical high-dimensional operators in canonical polyadic finite basis representation (CP-FBR)

Area of Science:

• Computational Quantum Chemistry
• Theoretical Chemistry
• Applied Mathematics

Background:

• High-dimensional functions are challenging to represent and compute with in quantum dynamics.
• Canonical Polyadic (CP) decomposition offers a compact representation but can be difficult to obtain directly from discrete data.
• Previous methods like Tucker sum-of-products-FBR exist but are less compact than CP.

Purpose of the Study:

• To introduce a simple, analytical (grid-free) method for obtaining a canonical polyadic (CP) representation of multidimensional functions from discrete data.
• To leverage auxiliary basis functions (finite basis representation - FBR) for this CP representation.
• To demonstrate the utility of the CP-FBR approach for high-dimensional quantum dynamics.

Main Methods:

• Utilized an initial, potentially unconverged, CP guess.
• Employed auxiliary basis functions (finite basis representation - FBR) in conjunction with the CP guess.
• Developed the CP-FBR expression as a compact, analytical representation.

Main Results:

• Successfully obtained an analytical CP representation from discrete data using the CP-FBR method.
• Demonstrated that the CP-FBR method requires a significantly coarser grid compared to traditional methods for dynamics.
• Showcased the method's applicability to bound systems of increasing dimensionality, including H2 (3D), HONO (6D), and CH4 (9D).

Conclusions:

• The CP-FBR method provides a compact and efficient means of representing multidimensional functions for quantum dynamics.
• The grid-free nature and coarser grid requirement significantly reduce computational demands.
• The basis functions can be interpolated for subsequent analysis, accommodating various initial conditions.