Direct unconstrained optimization of excited states in density functional theory (2501.10907v1)

Abstract: Orbital-optimized density functional theory (DFT) has emerged as an alternative to time-dependent (TD) DFT capable of describing difficult excited states with significant electron density redistribution, such as charge-transfer, Rydberg, and double-electron excitations. Here, a simple method is developed to solve the main problem of the excited-state optimization -- the variational collapse of the excited states onto the ground state. In this method, called variable-metric time-independent DFT (VM TIDFT), the electronic states are allowed to be nonorthogonal during the optimization but their orthogonality is gradually enforced with a continuous penalty function. With nonorthogonal electronic states, VM TIDFT can use molecular orbital coefficients as independent variables, which results in a closed-form analytical expression for the gradient and allows to employ any of the multiple unconstrained optimization algorithms that guarantees convergence of the excited-state optimization. Numerical tests on multiple molecular systems show that the variable-metric optimization of excited states performed with a preconditioned conjugate gradient algorithm is robust and produces accurate energies for well-behaved excitations and, unlike TDDFT, for more challenging charge-transfer and double-electron excitations.

• The paper introduces VM TIDFT, a novel method enabling direct unconstrained optimization of DFT excited states.
• It uses a continuous penalty function to enforce orthogonality and effectively prevent variational collapse.
• Numerical tests demonstrate that VM TIDFT reliably outperforms TDDFT for charge-transfer and double-electron excitations.

Insights into Direct Unconstrained Optimization of Excited States in Density Functional Theory

The research paper titled "Direct unconstrained optimization of excited states in density functional theory" presents a novel methodological advancement in the field of density functional theory (DFT) with a focus on excited state calculations. The authors, Hanh D. M. Pham and Rustam Z. Khaliullin from McGill University, propose an innovative approach named Variable-Metric Time-Independent DFT (VM TIDFT) that seeks to address significant challenges associated with excited state optimizations, particularly the problem of variational collapse.

Background and Motivation

Excited states are pivotal in various chemical processes and applications, such as photovoltaic materials, photocatalysis, and molecular electronic devices. Traditional time-dependent DFT (TDDFT) is widely used to paper excited states but suffers from limitations like difficulties in describing Rydberg, charge-transfer, and double-electron excitations—states characterized by notable electron density redistribution. Alternative methods, such as orbital-optimized (OO) DFT approaches, have been developed to overcome these TDDFT limitations. However, a critical challenge in OO DFT is avoiding the collapse of high-energy excited states into the ground state during optimization.

VM TIDFT Methodology

The VM TIDFT method stands out due to its simplified approach to excited-state optimization. It allows electronic states to be nonorthogonal during optimization, progressively enforcing orthogonality through a continuous penalty function. This feature enables the method to leverage molecular orbital coefficients as independent variables, providing a direct minimization pathway with closed-form analytical expressions for energy gradients. This approach markedly facilitates the use of efficient unconstrained optimization algorithms.

Critical to VM TIDFT is its interstate penalty term, which is designed to prevent variational collapse. It achieves this by computationally enforcing orthogonality between the electronic states, ensuring that the variational procedure locates the correct excited states rather than the lower-energy ground state. Notably, VM TIDFT proposes a practical algorithm that effectively minimizes a loss function encompassing both energy and orthogonality constraints.

Numerical Performance

The authors conducted comprehensive numerical tests across various molecular systems to demonstrate the robustness of the VM TIDFT approach. They show that when implemented with a preconditioned conjugate gradient algorithm, VM TIDFT offers accurate and stable optimization of excited states across different excitation challenges. For instance, results for charge-transfer and double-electron excitations yielded accurate energies, reflecting the method’s superiority over TDDFT in these cases. Furthermore, the paper highlights that the VM TIDFT significantly reduces challenges related to convergence and computational cost compared to existing methods.

Implications and Future Prospects

The development and application of VM TIDFT present several theoretical and computational implications. The method’s compatibility with a wide range of optimization algorithms enhances its adaptability and usefulness across various computational chemistry problems. The ability to directly optimize excited states in a robust, efficient manner opens opportunities in accurately modeling systems where traditional methods falter.

Looking forward, the extension of this methodology to incorporate spin-purification techniques, such as spin-purified open-shell singlet states, will likely improve accuracy further for complex excited states. Additionally, this approach presents a groundwork for refining coupled excited state and ground state calculations pivotal in computational spectroscopy and reaction dynamics studies.

In summary, VM TIDFT provides a simplified, efficient, and accurate method for the optimization of excited states, which holds promise for advancing the understanding and development of materials and reactions governed by complex electronic excited states.