## What is the variational principle of DFT?

For this functional a variational principle holds: **the ground-state energy is minimized by the ground-state charge density**. In this way, DFT exactly reduces the N-body problem to the determination of a 3-dimensional function n(r) which minimizes a functional E[n(r)].

**What do you mean by variational principle?**

In science and especially in mathematical studies, a variational principle is **one that enables a problem to be solved using calculus of variations**, which concerns finding functions that optimize the values of quantities that depend on those functions.

**Why do we use variational principle?**

The variational principle is a useful tool to have in our pocket because **it lets us leverage the Hamiltonians which we actually can solve to solve Hamiltonians which we can't**. The strategy of the variational principle is to use a problem we can solve to approximate a problem we can't.

**What is the variational principle of calculation?**

The variational principle states, quite simply, that the ground-state energy, E0, is always less than or equal to the expectation value of H calculated with the trial wavefunction: that is, **E0≤⟨ψ|H|ψ⟩**.

**Why is it called variational method?**

The term variational is used **because you pick the best q in Q** -- the term derives from the "calculus of variations", which deals with optimization problems that pick the best function (in this case, a distribution q).

**What is the first variational principle?**

The first variational principle of classical mechanics is the **principle of possible (virtual) displacements**, which was used as early as 1665 by G. Galilei. J. Bernoulli in 1717 was the first to grasp the generality of this principle and its usefulness for the solution of problems in statics.

**What is variational method differential equation?**

The variational equation of an autonomous system ˙x=f(x) at a fixed point (i.e. along a solution x(⋅)=x0) is **a linear system of differential equations with constant coefficients**, and, if f(⋅) is not varied, then the system is homogeneous for variations of the first order and "with quasi-polynomial right-hand side" for ...

**What is virtual work and variational principle?**

The principle of virtual work states that **in equilibrium the virtual work of the forces applied to a system is zero**. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint forces. This means the virtual work of the constraint forces must be zero as well.

**What is the meaning of variational problem?**

A variational problem with free (mobile) ends is **a problem in variational calculus in which the end points of the curve which gives the extremum may move along given manifolds**.

**What is variational inference in simple terms?**

Variational inference is **a method of approximating a conditional density of latent variables given observed variables**. It has also laid the foundation for Bayesian deep learning. There are many kinds of literature and documentation on this topic online.

## What is variational inference for beginners?

Variational Inference **aims to approximate the posterior with a “well behaved” distribution**. This means that integrals are computed such that the better the estimate, the more accurate the approximate inference will be .

**What are the advantages of variation principle formulation?**

The existence of variational principles for different physical theories is a great mystery in physics. A variational formulation of a physical problem can be useful because **we can use special solution methods, we can have a machinery to handle symmetries and a deeper insight into the structure of a theory**.

**What is the variational principle of least action?**

The Principle of Least Action now states that **among all the possible paths x(t) that connect the fixed endpoints, the actual path taken by the body, X(t), is the one that makes the action S minimal3**. Figure 5: Possible paths x(t). (t0 + t1).

**What is the principle of density functional theory?**

The underlying principle of DFT is that **the total energy of the system is a unique functional of the electron density** [1], hence it is unnecessary to compute the full many-body wave function of the system. However, the precise functional dependence of the energy on the density is not known.

**What is the variational method and why is it useful in quantum chemistry?**

The Variational Method is a mathematical method that is **used to approximately calculate the energy levels of difficult quantum systems**. It can also be used to approximate the energies of a solvable system and then obtain the accuracy of the method by comparing the known and approximated energies.

## References

- https://physics.stackexchange.com/questions/173308/is-density-functional-theory-a-mean-field-theory
- https://www.fisica.uniud.it/~giannozz/Didattica/MetNum/LectureNotes/metnum-cap2.pdf
- https://www.finishing.com/580/38.shtml
- https://www.sciencedirect.com/topics/engineering/circular-convolution
- https://www.eurekaselect.com/article/40744
- https://duluxpowders.com.au/wp-content/uploads/2020/11/Dulux_Powder_Technical_Advice_Brochure_Dry_Film_Thickness_WEB_09_20.pdf
- https://towardsdatascience.com/learn-discrete-fourier-transform-dft-9f7a2df4bfe9
- https://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l7.pdf
- https://painttoprotect.com/how-to-measure-wet-film-thickness/
- https://www.mathworks.com/help/images/fourier-transform.html
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/13%3A_Variational_Methods/13.01%3A_Variational_Principle
- https://www.geeksforgeeks.org/discrete-fourier-transform-and-its-inverse-using-c/
- https://en.wikibooks.org/wiki/Digital_Signal_Processing/Discrete_Fourier_Transform
- https://encyclopediaofmath.org/wiki/Variational_principles_of_classical_mechanics
- https://www.tutorialspoint.com/digital_signal_processing/dsp_discrete_time_frequency_transform.htm
- https://arxiv.org/abs/1201.3679
- https://en.wikipedia.org/wiki/Variational_principle
- https://www.portable-testers.com/coating-thickness-testers.html
- https://www.defelsko.com/coating-thickness/imo-pspc-90-10
- https://www.onropes.co.uk/coating-calculations
- https://www.researchgate.net/post/Which_is_best_software_for_Density_Functional_theory_calculation
- https://www.researchgate.net/post/What_are_the_differences_or_similarities_between_DFT_ground_state_energy_and_potential_energy_of_molecular_dynamics_simulation_MD
- https://www.damtp.cam.ac.uk/user/nsm10/PrincLeaAc.pdf
- https://en.wikipedia.org/wiki/Virtual_work
- https://www.linkedin.com/advice/1/how-do-you-compare-evaluate-different-3f
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/17%3A_Quantum_Calculations/Variational_Method
- http://www.rmki.kfki.hu/~vpet/Publ/VanNyi99a.pdf
- https://www.maplesoft.com/support/help/maple/view.aspx?path=SignalProcessing%2FDFT
- https://mattermodeling.stackexchange.com/questions/1513/what-are-the-types-of-dft
- https://www.section.io/engineering-education/implementation-of-2d-discrete-fourier-transform-of-square-functions-and-natural-images-in-matlab/
- https://stanford.edu/~jeffjar/statmech/lec5b.html
- https://www.ee-diary.com/p/dft-and-idft-calculator.html
- https://www.wavewalkerdsp.com/2022/03/16/dft-frequency-resolution-explained/
- https://stats.stackexchange.com/questions/340955/what-does-variational-mean
- https://dsp.stackexchange.com/questions/16586/difference-between-discrete-time-fourier-transform-and-discrete-fourier-transfor
- https://www.nobroker.in/forum/what-is-dft-in-painting/
- https://en.wikipedia.org/wiki/Fast_Fourier_transform
- http://vergil.chemistry.gatech.edu/courses/chem4681/background/node7.html
- https://en.wikipedia.org/wiki/Density_functional_theory
- https://pubs.acs.org/doi/10.1021/jp960669l
- https://arxiv.org/pdf/2112.03441
- https://www.researchgate.net/figure/Comparison-of-the-number-of-operations-for-ON-2-DFT-algorithms_tbl1_3601502
- https://erickson.academic.wlu.edu/files/courses2020/sigproc_s2020/readings/DFT_Intro.pdf
- https://royalsocietypublishing.org/doi/10.1098/rsta.2013.0270
- https://towardsdatascience.com/variational-inference-for-neural-networks-a4b5cf72b24
- https://www.dspguide.com/ch10/3.htm
- https://en.wikipedia.org/wiki/Discrete_Fourier_transform
- https://www.analog.com/media/en/technical-documentation/dsp-book/dsp_book_Ch9.pdf
- https://encyclopediaofmath.org/wiki/Variational_problem
- https://pubs.acs.org/doi/10.1021/acs.jctc.7b01252
- https://www2.icp.uni-stuttgart.de/~icp/mediawiki/images/b/b8/DFT.pdf
- https://www.cs.cmu.edu/afs/andrew/scs/cs/15-463/2001/pub/www/notes/fourier/fourier.pdf
- https://www.physik.uzh.ch/local/teaching/SPI301/LV-2013-Help/lvanlsconcepts.chm/Power_Spect.html
- https://www.nature.com/articles/s41467-020-19093-1
- https://www.defelsko.com/product-categories/coating-thickness
- https://pubs.acs.org/doi/10.1021/ed5004788
- https://leimao.github.io/article/Introduction-to-Variational-Inference/
- https://www.corrosionpedia.com/2/3068/environments/marine/the-impact-of-minimum-maximum-dft-values-on-coating-performance
- https://arxiv.org/html/math/0302212
- https://www.corrosionpedia.com/definition/1249/dry-film-thickness-dft
- https://www.azonano.com/article.aspx?ArticleID=6372
- https://www.ni.com/docs/en-US/bundle/diadem/page/genmaths/genmaths/calc_fouriertransform.htm
- https://www.diffzy.com/article/difference-between-fft-and-dft-862
- https://encyclopediaofmath.org/wiki/Variational_equations
- https://math.stackexchange.com/questions/30464/what-is-the-difference-between-the-discrete-fourier-transform-and-the-fast-fouri