On the performance of QTP functionals applied to second-order response properties II: Dynamic polarizability and long-range C$_6$ coefficients

This study extends the evaluation of Quantum Theory Project (QTP) functionals to dynamic polarizabilities and long-range C$_6$ dispersion coefficients, identifying TPSS0 and QTP01 as the top performers for dynamic polarizabilities and QTP01 and LC-QTP as the best within the QTP family for C$_6$ coefficients among 25 tested exchange-correlation functionals.

The Gist

Imagine you are trying to predict how a molecule behaves when you shine different colored lights on it, or how two molecules "feel" each other's presence from far away without touching. This is the job of Quantum Theory Project (QTP) functionals, which are like sophisticated rulebooks that tell computers how to calculate these behaviors.

This paper is the second chapter in a story about testing these rulebooks. In the first chapter, the authors tested how well these rules worked for things that don't change with time (static properties). In this chapter, they put the rules to the test with moving targets: things that change when the "light" (or electric field) changes speed.

Here is a breakdown of what they did and what they found, using simple analogies:

1. The Two Big Tests

The authors put 25 different rulebooks (called "functionals") through two main challenges:

• Challenge A: The "Jiggling" Test (Dynamic Polarizability)

  • The Analogy: Imagine a molecule is a rubber ball. If you poke it gently, it squishes a little. If you poke it fast, it squishes differently. "Dynamic polarizability" measures how much the molecule squishes (changes its shape) when hit by an electric field oscillating at different speeds (colors of light).
  • The Setup: They hit the molecules with five different "colors" of light, ranging from a slow red glow to a very fast, high-energy ultraviolet flash.
  • The Goal: See which rulebook predicts the squishiness most accurately compared to the "Gold Standard" (a super-accurate but very expensive computer method called CC3).

• Challenge B: The "Long-Range Hug" Test (C6 Coefficients)

  • The Analogy: Imagine two people standing far apart in a dark room. They can't see each other, but they can feel a faint magnetic pull or a "hug" from a distance. In chemistry, this is called the van der Waals force. The C6 coefficient is a number that measures how strong this invisible hug is.
  • The Goal: See which rulebook can best predict the strength of this long-distance hug by comparing their math to real-world experimental data.

2. The Results: Who Won the Race?

For the "Jiggling" Test (Dynamic Polarizability)

• The Gold Standard: The most accurate method (EOM-CCSD) was almost perfect for the slower lights, but when the light got very fast (the highest frequency), it started to stumble a bit. This is like a runner who is great at a slow jog but trips when sprinting.
• The Rulebook Winners:
  • TPSS0 and QTP01: These two rulebooks were the clear champions. They handled the "squishiness" at all speeds better than almost everyone else.
  • The "High-Speed" Problem: Almost all rulebooks struggled when the light was the fastest (ultraviolet). It's as if the rulebooks didn't have a good map for the "high-energy" part of the molecule's behavior.
  • The "Pole" Check: The authors also checked if the rulebooks could predict the "resonance points" (like the exact note a guitar string vibrates at). Most rulebooks got the first few notes right, but they all missed the highest, sharpest notes. However, the QTP family (specifically QTP01) got closer to the truth than the others.

For the "Long-Range Hug" Test (C6 Coefficients)

• The Surprise Winner: O3LYP took the top spot, predicting the strength of the molecular "hug" with incredible accuracy (only 3.3% off from reality).
• The Tight Pack: The top 11 rulebooks were all very close to each other, like runners in a pack where the difference between 1st and 11th place is a fraction of a second.
• The QTP Family: Within the specific family of QTP rulebooks, LC-QTP and QTP01 were the best performers, beating out their older siblings.
• The Loser: The oldest, simplest rulebook (SVWN5) was way off, predicting the hug was much weaker than it actually was.

3. The Big Takeaway

Think of these functionals as different types of navigation apps for a chemical journey.

• Some apps (like the older, simpler ones) get you lost in the city center.
• Some apps (like the "Range-Separated" ones) are great for long highway drives but get confused in the city.
• The QTP family (especially QTP01 and LC-QTP) and TPSS0 are like the new, high-tech GPS systems. They handle both the slow, steady cruising (static properties) and the fast, tricky turns (dynamic properties) better than the competition.

In short: If you want to know how a molecule reacts to changing light or how it interacts with a neighbor from far away, the authors recommend using the QTP01, LC-QTP, or TPSS0 rulebooks. They are the most reliable guides for these specific chemical mysteries.

Technical Summary

1. Problem Statement

This study addresses the accuracy of Density Functional Theory (DFT) functionals in predicting frequency-dependent second-order response properties, specifically dynamic polarizabilities and long-range dispersion ($C_6$) coefficients.

While the first paper in this series demonstrated the success of Quantum Theory Project (QTP) functionals for static properties, this work investigates their performance under oscillating electric fields. The core challenge lies in the "Devil's Triangle" of Kohn–Sham DFT: self-interaction error, lack of integer discontinuity, and incorrect one-particle spectra. The authors aim to determine if QTP functionals, which are derived from Correlated Orbital Theory (COT) and mapped to Coupled-Cluster (CC) theory, can overcome these limitations to provide accurate frequency-dependent responses comparable to high-level wavefunction methods like EOM-CCSD and LR-CC3.

2. Methodology

Computational Framework:

• Functionals Investigated: A total of 25 exchange-correlation (XC) functionals were tested, spanning all rungs of Jacob's Ladder (LDA, GGA, meta-GGA, global hybrids, and range-separated hybrids). This includes the QTP family (QTP00, QTP01, QTP02, LC-QTP) and standard functionals (e.g., B3LYP, PBE0, CAM-B3LYP, TPSS0).
• Reference Methods:
  • Dynamic Polarizability: Results were benchmarked against Linear Response CC3 (LR-CC3) calculations using the aug-cc-pVTZ basis set. EOM-CCSD and Hartree-Fock (HF) were also computed for comparison.
  • $C_6$ Coefficients: Benchmarked against experimental data.
• Software:
  • CC3 and EOM-CCSD: CFOUR and ACES II packages.
  • DFT: PySCF (v2.7.0) with the LibXC library.
• System Sets:
  • Dynamic Polarizability: 13 molecules (e.g., HF, H$_2$O, CO$_2$, SO$_2$) at experimental geometries.
  • $C_6$ Coefficients: 21 molecules (e.g., C$_2$H$_2$, CO, N$_2$, H$_2$O) with geometries from Cheng and Verstraelen.
• Perturbation Frequencies: Dynamic polarizabilities were calculated at five wavelengths: 632.99 nm, 594.10 nm, 543.52 nm, 514.50 nm, and 325.13 nm.
• $C_6$ Calculation: Performed using the Casimir-Polder equation, integrating imaginary-frequency-dependent polarizabilities via Gauss-Legendre quadrature.

3. Key Contributions

1. Systematic Benchmarking of QTP Functionals: The study provides a comprehensive evaluation of the QTP family (derived from COT) against a broad spectrum of standard XC functionals for dynamic properties.
2. Identification of Top Performers: It identifies TPSS0 and QTP01 as the most robust functionals for dynamic polarizabilities across a wide frequency range, and O3LYP (along with QTP01/LC-QTP) for $C_6$ coefficients.
3. Analysis of Pole Structure: The authors analyzed the pole structure of the CO molecule's response function, assessing how well different functionals reproduce excitation energies and residues compared to EOM-CCSD.
4. Frequency Dependence Analysis: The work highlights a specific limitation where most functionals (including high-level EOM-CCSD) degrade in accuracy at the highest frequency (325.13 nm), likely due to poor description of high-lying Rydberg states.

4. Key Results

A. Dynamic Polarizabilities:

• Reference Accuracy: EOM-CCSD results were extremely close to LR-CC3 (MAD 0.09–0.10 a.u.) for the first four frequencies but showed a significant deviation (MAD 0.22 a.u.) at the highest frequency (325.13 nm).
• Functional Performance:
  • Best Overall: TPSS0 (meta-GGA hybrid) showed the best overall performance (MAD 0.23–0.24 a.u.) across all frequencies.
  • QTP Performance: QTP01 was the second-best overall (MAD 0.28–0.29 a.u. for lower frequencies; 0.40 a.u. at highest frequency). LC-QTP and QTP02 also performed well. QTP00 was the weakest in the QTP family.
  • Trends: Functionals including second-order ingredients (density Laplacian/kinetic energy density, i.e., meta-GGAs) generally outperformed pure GGAs and LDA. Range-separated hybrids (RSH) like LC-$\omega$PBE performed well, but none surpassed TPSS0.
  • High-Frequency Error: All methods showed increased error at 325.13 nm, attributed to the difficulty in modeling high-lying Rydberg and valence states.
• Pole Structure (CO Molecule):
  • Most functionals (including QTP variants) reproduced the position of the first pole well.
  • The second pole was systematically shifted by DFT approximations.
  • QTP00, QTP01, and TPSS0 showed the best qualitative agreement with the reference pole pattern, particularly in capturing the "plateau" region of polarizability between 0.32 and 0.41 a.u., which other functionals missed. CAM-B3LYP failed to reproduce pole positions and magnitudes accurately.

B. Long-Range $C_6$ Coefficients:

• Top Performer: O3LYP (GGA-hybrid) achieved the lowest error (3.30% vs. experiment).
• Top Tier: The top 11 functionals all had errors below 4%. This group included global hybrids (TPSS0, PBE0) and RSHs (LC-$\omega$PBE).
• QTP Performance: LC-QTP and QTP01 tied for the best QTP results (3.60% error), slightly outperforming CAM-B3LYP (3.71%). QTP00 was the worst QTP functional (4.83% error).
• Systematic Bias: Most functionals underestimated $C_6$ coefficients. LC-QTP had the smallest signed error (-0.06%), while SVWN5 had the largest (-14.98%).

5. Significance

This paper establishes that QTP functionals, particularly QTP01 and LC-QTP, are highly competitive with, and in some cases superior to, standard hybrid and range-separated functionals for predicting dynamic response properties.

• Validation of COT Mapping: The success of QTP functionals supports the Correlated Orbital Theory approach of mapping KS-DFT onto CC theory, suggesting it effectively mitigates self-interaction errors and improves the description of orbital eigenvalues and response properties.
• Practical Utility: The identification of TPSS0 and QTP01 as top performers offers computational chemists reliable, cost-effective alternatives to expensive CC3 calculations for dynamic polarizabilities and dispersion coefficients.
• Limitations Identified: The study clarifies that while these functionals are robust, they (like EOM-CCSD) struggle at very high frequencies (UV region), indicating a need for further development in describing high-lying excited states within the DFT framework.

In conclusion, this work solidifies the QTP family as a leading candidate for accurate second-order response property calculations, bridging the gap between the accuracy of coupled-cluster methods and the efficiency of DFT.