Theory Flashcards
What are three types of approximations involved in a DFT calculation?
- One is conceptual, and concerns the interpretation of KS eigenvalues and orbitals as physical energies and wave functions → DFT is not a many-body theory, but a mean-field theory.
- The second type is numerical, and concerns methods for actually solving the SE. A main aspect here is the selection of suitable basis functions.
- The third type of approximation involves constructing an expression for the unknown xc functional Exc[n], which contains all many-body aspects of the problem.
What is the idea behind the pseudopotential (PP) approach, used in DFT calculations?
- Chemical binding in molecules and solids is dominated by the outer (valence) electrons of ach atom
- The inner (core) electrons retain an atomic-like configuration
- Therefore, only the valence density is determined self-consistently, while core electrons are treated by atomic calculations
- Original KS eq: ν[n]=νext+νH[n]+νxc[n] → in the PP approach, the Hartree and xc terms are evaluated only for the valence density nv, and the core electron are accounted by replacing the external potential by a pseudo potential → ν[nv]=νextPP+νH[nv]+νxc[nv]
- Reduces the number of electrons to be treated explicitly → makes it possible to perform DFT calculation for large systems
What is the idea behind the local-density approximation (LDA) functionals?
Historically, LDA is the most important type of approximation
- The non-interacting (single-body) kinetic energy in an inhomogeneous system is approximated locally to a homogeneous system and intgrated over space → much inferior to the exact treatment of kinetic energy in terms of orbitals
- The exchange energy is also approximated here to a locally homogeneous system, but an expression for the correlaton energy in a homogeneous electron is not known → Ec cannot be approx.
- Still, LDA very popular because it presents a systematic error cancellation → it underestimates Ec and overestimates Ex
What is the most used correction to local-density approx. LDA functionals (concept and examples)?
- LDA assumes loccally homogeneous system → Any real system is spatially inhomogeneous, i.e., it has a spatially varying density n(r) → the most obvious correction is to try to add information on the rate of this variation → generalized-gradient approximations (GGAs)
- Nowadays the most popular GGAs are: PBE (denoting the functional proposed in 1996 by Perdew, Burke and Ernzerhof) in physics; and BLYP (denoting the combination of Becke’s 1988 exchange functional with the 1988 correlation functional of Lee, Yang and Parr) in chemistry.
- Beyond-GGA corrections include: B3LYP, with three-parameter hybrid functional for correlation; self-interaction correction PY-SIC; optimized effective potential (OEP/OPM); average-density approximation (ADA); weighted-density approximation (WDA). Last two are too expensive and not worth it for small systems (LDA prefered).
Why it is not so relevant that the energies calculated match perfectly the experimental XAS spectrum?
- In poliatomic systems, it is known that too many approximations are made for the calculations, therefore, it is expected that the calculated absolute energies won’t match the experiment
- For simpler systems, like an atomic spectrum, a certain accuracy is expected
- For calculation of frequescies in IR spectra it also expected some accuracy, as the models are based on classical physics (bond = spring → Hooke’s Law)
what are the differences between DFT and CC calculations?
Approach:
* DFT is based on electron density and uses functionals to approximate exchange-correlation energy.
* CC methods solve the Schrödinger equation directly and consider electron correlation explicitly.
Accuracy:
* DFT can provide reasonable accuracy but may fail in specific scenarios involving strong correlations or dispersion forces.
* CC methods typically yield very high accuracy, often considered benchmark quality.
Computational Cost:
* DFT is generally faster and more efficient, making it suitable for large systems.
* CC methods are computationally intensive and less feasible for large-scale calculations.
why is TD-DFT better for calculating K edges than L edges? And what about M and N edges?
- K-edge transitions involve excitations from the 1s core orbital, which is more localized and has a stronger interaction with the incoming X-ray photon. This results in well-defined transitions that can be effectively modeled using TD-DFT, which allows for the restriction of the excitation window to specific core orbitals.
- L-edge transitions typically involve excitations from 2s and 2p orbitals, which can lead to more complex interactions, including spin-orbit coupling effects. These transitions are often less straightforward to model accurately with TD-DFT due to their dependence on a larger number of orbitals and the need to account for additional electron correlation effects.
- The M and N edges may require relativistic treatment to accurately model the interactions, especially for heavier elements. For N edges, the challenges are even more pronounced due to the weaker transitions and potential overlap with valence states, which can lead to artifacts in the spectra if not handled correctly.
What is the self-interaction error (SIE) that appears in DFT calculations, and why does it particularly affect d-electrons?
In a many-electron system, each electron interacts with every other electron. This interaction can be described using Coulomb’s law, which states that the potential energy V between two point charges (electrons) is given by:
V=(k . q1 . q2)/r
where k is a constant, q1 and q2 are the charges (in this case, the charge of an electron), and r is the distance between them
When we consider an electron interacting with itself, it should ideally not contribute to its own interaction energy. However, in DFT, the approximate functionals used do not completely eliminate this self-interaction. Thus, the energy contribution from an electron interacting with itself is not perfectly canceled out. The SIE causes localized d-electrons to be spread out over a larger region than they should be.
PS: coupled cluster calculations mitigate the self-interaction error by employing a wave function-based approach that explicitly accounts for electron correlation, avoiding reliance on approximate functionals.
What is the exchange-correlation energy and how is it connected to an unaccuracy of DFT calculations?
- Exchange effects arise from the indistinguishable nature of electrons, which are fermions. Due to the Pauli exclusion principle, no two electrons can occupy the same quantum state simultaneously. When two electrons are exchanged, the overall wave function of the system must change sign, leading to an “exchange energy” that reflects this antisymmetry.
- Correlation arises because electrons do not move independently; their movements are correlated due to their mutual interactions. This correlation leads to a reduction in the total energy of the system compared to what would be predicted by a mean-field approach.
- Exc is defined as the difference between the exact energy of the system and the classical energy calculated using the electron density
Exc[ρ] = E[ρ] - (T[ρ] + Vee[ρ])
E[ρ]: total energy os the system
T[ρ]: kinetic energy of the non-interacting system
Vee[ρ]: classical electron-electron interaction energy
= (k . q1 . q2)/r - Classical models treat electrons as distinguishable point particles, whereas quantum mechanics treats them as indistinguishable entities described by probabilistic wave functions
- In DFT, the exchange-correlation energy is calculated based on the assumption that the electron density is uniform throughout the system (LDA). This leads to an underestimation of the exchange energy and an overestimation of the correlation energy
what are spin-restricted and spin-unrestricted calculations?
- In spin-restricted calculations, the spatial orbitals and the occupation numbers for spin-up (α) and spin-down (β) electrons are identical. This means that both types of electrons occupy the same spatial orbitals, which is appropriate for systems with a closed-shell electronic configuration
ψα(r)=ψβ(r) - In spin-unrestricted calculations, separate sets of orbitals are allowed for spin-up and spin-down electrons. This means that the spatial orbitals can differ between the two spins, which is particularly useful for systems with an odd number of electrons or unpaired electrons (e.g., radicals)
ψα(r)≠ψβ(r)
what are ab initio calculations and how do they differ from other types?
- ab initio calculations aim to predict molecular structures and properties based on fundamental principles of quantum mechanics, without relying on empirical parameters. These methods are grounded in the principles of quantum mechanics and solve the Schrödinger equation for a system from first principles
- For example, DFT is an ab initio method that focuses on electron density rather than wave functions. On the other hand, semi-empirical methods simplify calculations by using parameters derived from experimental measurements. Plus, molecular mechanics uses classical physics principles to model molecular systems, treating atoms as classical particles connected by bonds
What are multireference calculations and when are they applied?
- Multireference calculations account for multiple configurations (references) that contribute to one electronic state. For example, in CCSD or CCSD(T) calculations, single, double or triple excitations are considered to find possible configurations contributing to a certain state, e.g. let’s consider we have a ground state that is composed of
65% 3d5 4s0
25 % 3d4 4s1 → single excitation
10% 3d3 4s1 4p1 → double excitation - In single reference calculations, only one configuration is considered for each state. On the other hand, if one wants to calculate spectra with multiref., then mutilple config. are considered for ground and excited states → multireference calculations are much more expensive computationally than TD-DFT
- Multiref. methods are typically necessary for systems exhibiting significant static correlation. Static correlation is associated with near-degeneracy of electronic states. Multiref. character might be significant, e.g., in systems with terminal oxo ligands or double bonds.
What is the main concept of DFT calculaitons?
- DFT maps a many-body problem onto a single-body problem by taking the particle density ρ(r) as key variabe for calculating all other observables
ρ(r) → Ψ(r1, r2, …, rN) → V(r)
knowledge of ρ implies the knowledge of Ψ and all observables - The electron density ρ(r) is a function of spatial coordinates, while the functional F[ρ] operates on this entire function to produce a single number F, that could be the energy or other physical properties of the system. The most simple approx. would be the LDA, but more sophisticated aproaches are the GGA functionals like PBE or B3LYP (with three-parameter hybrid functional for correlation).
- Basis sets are collections of functions used to represent molecular orbitals and electronic wavefunctions in computational calculations. They serve as building blocks for constructing more complex electronic structures, examples are the most simple Gaussian-type orbitals (GTOs) or the ZORA basis sets that are optimized for relativistic calculations.
What are the T1 and D1 diagnostics?
- The T1 and D1 diagnostics are widely used tools in quantum chemical calculations, particularly in coupled cluster (CC) methods, to assess the quality of single-reference electron correlation methods and determine the multireference character of molecular systems
- The T1 diagnostic provides a measure of the importance of single excitations in the coupled cluster wave function. A high T1 value indicates significant single excitation character, which may suggest the need for a multireference treatment.
- The D1 diagnostic is related to the T1 diagnostic but provides a different measure of the coupled cluster wave function quality
- In our systems, values of T1 ≥ 0.045 and D1 ≥ 0.120 were proposed as a gauge to predict the possible need to employ multireference wave function-based methods
what is the difference between the electronic and the zero-point corrected energies?
Electronic Energy:
* Represents the energy of electrons in their ground state configuration at 0 K.
* Obtained directly from solving the electronic Schrödinger equation.
* Does not account for nuclear motion or vibrational effects.
Zero-Point Corrected Energy:
* Includes the electronic energy plus the zero-point vibrational energy (ZPVE).
* ZPVE is the lowest possible vibrational energy a molecule can have, even at absolute zero temperature.
* Accounts for the quantum mechanical effect that molecules retain some vibrational motion even at 0 K.
The zero-point corrected energy is generally considered more accurate for comparing molecular energies, as it includes the contribution from zero-point vibrations.
This correction is particularly important for:
* Comparing energies of molecules with different numbers of atoms.
* Calculating accurate reaction energies and barrier heights.
* Studying isotope effects, as ZPVE differs between isotopes (e.g., hydrogen vs. deuterium).
typical energy separation between vibronic levels in a 5-atom molecule in its electronic ground state is approximately 0.1 to 0.5 eV.
What is the difference between CCSD(T) and CASSCF methods?
Theoretical Approach:
CCSD(T): Uses coupled cluster theory with single, double, and perturbative triple excitations.
CASSCF: Employs a multiconfigurational approach using a linear combination of configuration state functions.
Electron Correlation:
CCSD(T): Primarily captures dynamic correlation, excelling for stable closed-shell molecules.
CASSCF: Focuses on static correlation, suitable for systems with quasi-degenerate ground states.
Reference State:
CCSD(T): Typically uses a single reference (usually Hartree-Fock).
CASSCF: Multi-configurational, allowing for multiple reference states.
Accuracy:
CCSD(T): Often considered the “gold standard” for single-reference problems, capable of achieving chemical accuracy (< 1 kcal/mol).
CASSCF: Provides a qualitatively correct description of multireference systems but may lack dynamic correlation.
Applicability:
CCSD(T): Highly accurate for small to medium-sized molecules, especially single-reference problems.
CASSCF: Well-suited for multi-reference problems, excited states, and transition metal complexes.
Computational Cost:
CCSD(T): Scales as O(N^7) with system size, limiting its application to larger systems.
CASSCF: Scales factorially with the size of the active space but has favorable scaling with basis set size for a fixed active space.
In summary, CCSD(T) is preferred for high-accuracy calculations of single-reference systems, while CASSCF is better suited for multi-reference problems and systems with strong static correlation.
what is the difference between complete and restricted active space calculations?
Complete Active Space (CAS) and Restricted Active Space (RAS) calculations are both multireference methods used in quantum chemistry, but they differ in how they handle electron excitations:
Orbital Classification:
CAS: Divides orbitals into core, active, and virtual spaces.
RAS: Divides orbitals into more subspaces, typically RAS1, RAS2, and RAS3.
Electron Excitations:
CAS: Performs a full configuration interaction (FCI) calculation within the active space, allowing all possible electron configurations.
RAS: Imposes restrictions on electron excitations between subspaces, typically allowing only limited excitations from RAS1 to RAS3.
Flexibility:
CAS: More rigorous but limited to smaller active spaces due to computational cost.
RAS: More flexible, allowing larger active spaces by restricting excitations.
Computational Cost:
CAS: Generally more computationally expensive for the same number of active orbitals.
RAS: Can handle larger systems with lower computational cost.
Application:
CAS: Often used for smaller systems or when a full treatment of static correlation is necessary.
RAS: Useful for larger systems or when a balance between accuracy and computational cost is needed.
Summarize the conceptual difference between Hartree-Fock and post-Hartree-Fock methods.
Hartree-Fock Method:
Approximates the multi-electron wave function as a single Slater determinant.
Treats electron-electron interactions in an averaged way.
Provides a variational solution, optimizing single-particle wave functions.
Serves as the starting point for more advanced calculations.
Post-Hartree-Fock Methods:
Developed to improve upon the HF method by including electron correlation.
Aim to capture the instantaneous interactions between electrons.
Generally provide more accurate results than HF, but at higher computational cost.
Include techniques such as Configuration Interaction (CI), Coupled Cluster (CC), and Møller-Plesset perturbation theory.
What is the idea behind Hartree equations?
- Simplification: They break down a multi-electron wave function into a set of one-electron wave functions, called molecular orbitals.
- Approximation of electron-electron interactions: Instead of calculating repulsion for all electron pairs, the Hartree method approximates the repulsion between each electron and an average field of all other electrons.
- Hartree Product: The wave function is expressed as a product of one-electron wave functions (spin-orbitals), each depending on three spatial coordinates and one spin coordinate.
- Limitations: The Hartree product has a significant flaw in that it doesn’t account for the indistinguishability of electrons and doesn’t satisfy the Pauli exclusion principle.
- Iterative solution: Due to the nonlinearities introduced by the approximation, the equations are solved using iterative methods, giving rise to the name “self-consistent field method”.
What do the basis set used in your calculations mean?
A basis set is a set of functions that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the quantum chemistry community; plane waves which are typically used within the solid state community, or real-space approaches. Polarization functions are added to describe polarization of the electron density of the atom in molecules.
aug-cc-pVTZ(-PP)
The ‘cc-p’, stands for ‘correlation-consistent polarized’ and the ‘V’ indicates that only basis sets for the valence orbitals are of multiple-zeta quality. The aug- prefix is added if diffuse functions are included in the basis. The (-PP) is for pseudo-potential, only used for the metal orbitals.
ZORA-def2-TZVP
ZORA: zero-order regular approximation. Relativistically recontracted version of Valence triple-zeta polarization. def2 is a type of split-valence basis sets (valence orbitals are composed of two basis functions each)
SARC-ZORA-TZVP
SARC: segmented all-electron relativistically contracted.
zeta, ζ, was commonly used to represent the exponent of an STO basis function. Slater-type orbitals (STOs) or Slater-type functions (STFs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method.
How is dft and the hartree equations correlated?
The Hartree equations and Density Functional Theory (DFT) are correlated as both address the quantum many-body problem by simplifying interactions among electrons, but they do so differently:
1. Hartree Equations: These approximate the many-electron wave function by treating electron interactions as an average field (mean-field approximation). The electron-electron Coulomb repulsion is included via the Hartree potential, but exchange and correlation effects are neglected.
2. DFT: DFT reformulates the problem using electron density (n(r)) instead of wave functions. It includes the Hartree term for electron-electron repulsion but adds an exchange-correlation functional (VXC) to account for many-body effects, improving accuracy.
Both use self-consistent iterative methods to solve their equations, but DFT is more comprehensive due to its inclusion of exchange-correlation effects.
