Perturbation Theory - Science topic

In the Moller-Plesset Perturbation Theory, the second order correction to the ground state energy can be divided into a same-spin and an opposite-spin terms by integrating the spin out. This allows to get integrals using only space orbitals, just as specified on the psi4 website here: https://psicode.org/psi4manual/master/dfmp2.html

I was able to get the expresion for opposite spins. However, when trying to perform the integration for same spins, I haven't been able to get rid of one of the terms and get the expresion I want. This happens because all four orbitals have the same spin function, so every term should mantain after the integration.

One of the main problems of semigroup theory for linear operators is to decide whether a concrete operator is the generator of a semigroup and how this semigroup is represented.

One idea is to write complicated operators, as a sum of simple operators. For this reason, perturbation theory has become one of the most important topics in semigroup theory. My question is about the multi-perturbed semigroups or multiple perturbation of semigroups in a Banach space. I need a recurrent formula for a semigroup perturbed by multiple (several, i.e. more than two) bounded (in general unbounded) linear operators. I have searched for it, but only found a simple case, called the Dyson-Phillips series for a semigroup generated by A₀+A₁. How can we find the generalisation of this formula for a semigroup generated by A₀+A₁+...+A_n for a fixed natural n? Many thanks in advance. I am looking forward to your suggestions and recommendations on this topic.

I'm working with molecular simulation of systems composed of nonspherical particles and currently interested in calculating the first- and second-order perturbation coefficients and comparing the results with the spherical case. These coefficents are related to the high-temperature power series expansion steming from Zwanzig's perturbation theory. I've come across an article ( ) that relates the ratio of these coefficients with the low-density limit. In other words:

a2/a1 = 0.5

or, in a most general case:

a_i/a_i-1 = 1/i

where a_i represent the perturbation coefficients and i is an integer number greater than or equal to 2. Does anyone know where this "perturbation ratio" comes from?

Feynman's parton model (as presented by, for example, W.-Y. P. Hwang, 1992, enclosed) seems to bridge both conceptions, but they do come across as mutually exclusive theories. The S-matrix program, which goes back to Wheeler and Heisenberg (see D. Bombardelli's Lectures on S-matrices and integrability, 2016, enclosed), is promising because - unlike parton theories - it does not make use of perturbation theory or other mathematically flawed procedures (cf. Dirac's criticism of QFT in the latter half of his life).

Needless to say, we do not question the usefulness of the quark hypothesis to classify the zoo of unstable particles (Particle Data Group), nor the massive investment to arrive at the precise measurements involved in the study of high-energy reactions (as synthesized in the Annual Reviews of the Particle Data Group), but the award of the Nobel Prize of Physics to CERN researchers Carlo Rubbia and Simon Van der Meer (1984), or Englert and Higgs (2013), seems to award 'smoking gun physics' only, rather than providing any ontological proof for virtual particles.

To trigger the discussion, we attach our own opinion. For a concise original/eminent opinion on this issue, we refer to Feynman's discussion of high-energy reactions involving kaons (https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S5), in which Feynman (writing in the early 1960s and much aware of the new law of conservation of strangeness as presented by Gell-Man, Pais and Nishijima) seems to favor a mathematical concept of strangeness or, more to the point, a property of particles rather than an existential/ontological concept. Our own views on the issue are summarized in (see the Annexes for our approach of modeling reactions involving kaons).

Some months ago, I solved a problem that required an approximate solution to the non-linear, second order, ordinary differential equation:

x" = x^2,                                                                          (1)

but I have since forgotten how to approach this, and am a little confused as to which method, or methods, I should use.

I seem to remember writing the equation as:

x" = (x + k)^2,                                                                 (2)

where k is a very small perturbation from the trajectory.

The aim is to determine whether or not the trajectory, orbit, or path, is stable, and I am also aware that one can even write u as a truncated series expansion, depending on the accuracy of the solution required, as follows:

x = x_0 + kx_1 + (k^2)x_2 + (k^3)x_3 + . . .

and

d(x)/dt = d(x_0 + kx_1 + (k^2)x_2 + (k^3)x_3 + . . .)/dt

where at t = 0, x(0) = a,

and then substitute at least one of these expansions into equation (1).

I have made many attempts to solve this problem recently (including multiplying both sides of equation (1) by the  derivative of x with respect to t, and then integrating - which only partially helps, as I only then get half way to the solution – and I have also attempted using matrices,  the Jacobian, its associated eigenvalues, and eigenvectors amongst other techniques), but I am really getting nowhere, and I cannot even find a remotely similar solution in any of my books, nor anywhere at all on the Internet.

I also seem to remember that the original question, up to six months ago, possibly required calculation of the zeros, prior to solution (which I think, in this case would be the two repeated values of x = - k, unless this was related to another question), so I wondered if anyone could suggest one or more appropriate methods of solution to adopt – along with the name of equation (1) and any texts that may highlight this particular ODE, along with further details of solution.  I also think that it is related to the ODE, known as the equation of path, in Newtonian Mechanics, perturbation theory and non-linear dynamics, as follows:

u” + u = a + bu^2,                                                         (3)

but would certainly appreciate any suggestions for equation (1).

Thanks.  SWM

The mathematics behind the inverse of large sparse matrices is very interesting and widely used in several fields. Sometimes, It is required to find the inverse of these kinds of matrices. However, finding the same is computationally costly. I want to know, the related research, what happens when a single entry (or a few entries) are perturbed in the original matrix then how much it will affect the entries of inverse of the matrix.

I am interesting in the magnetic anisotropy energy (MAE) due to spin-orbit coupling.

According to the force theorem, the MAE can be calculated as the difference in band energies for the two magnetization directions. On the other hand, we can directly compare the total energies corresponding to these directions. Doing VASP calculations I found that the two approaches give different results, even qualitatively. Why?

In gaussian, NBO analysis is done but I am not confidently able to decide the number of interacting molecules under the heading of Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis.

Please help if possible

Hello, I'm studying a type of chemical structures with intramolecular hydrogen bonds using gaussian09. I optimized the geometry of the said structures with M062X 6-31+g(d), and they seem to display strong hydrogen bonds (based of the distances). I want to complement this with the stabilization energies E(2) from Second Order Perturbation Theory in Natural Bond Orbital population analysis. From what I understand I should have a line in the output similar to LP ( 1) O a /***. BD*( 1) O b - H c (considering a hydrogen bond between two carboxyl groups) corresponding to the interaction nOa -> σ*Ob-Hc, however, I can't find any line with BD*( 1) O - H in the output files. I then found the lines LP ( 1) O a /***. LP*( 1) H c, that seem to have the E(2) values I should be expecting, what does this mean? Can somebody help please?

How we can perturb an abstract model.?Like if we have a system of equations $dx/dt=f(x,y), dy/dt=g(x,y), x(0)=x_0, y(0)=y_0$. If we wish to perturb the given system, so in how many ways one can do that? Like what I saw that we can do it by adding perturbation term(s) to right side of the above system, varying the initial conditions, or adding some singularity terms e.g multiplying epsilon to left side of any of above equations.

I carried out an NBO analysis though gaussian09 on the attached molecule, and the second order perturbation theory analysis indicated that the p-orbital on the top atom was donating electron density to the horisontal antibonding orbital shown.

The stabilisation energy for this interaction was ~10 kCal/mol-1, so presuming that this interaction was hyperconjugation, i visualised the two MOs in gaussview, but found that despite good overlap, that the MOs were out of phase. I was under the impression that this didnt make a difference as this would just be representing the unoccupied destabilising interaction, similar to this diagram (https://imgur.com/a/6CIw9Tu). However, I haven't been able to swap the phase when visualising the MO, so i dont know if i've just misunderstood.

Any help would be appreciated.

In most articles I have seen that derivation is only done to the first order correction in the energy eigenvalues for the case of time independent degenerate perturbation.

In the Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis I am not able to identify the transition (whether σ -σ* or π -π*) between donor NBO and acceptor NBO. Kindly help me in this regard. I am attaching here nbo output of water molecule.

I am looking for equations to do this calculation and for a book to cite it in my upcoming papers.

Thanks in advance.

Non-Linear waves propagating in plasmas

Time independent perturbation theory is sometime also called stationary perturbation theory.

Basically perturbation actually means small disturbance and we all know that such disturbance occurs in some time. So my question is how it is possible to called it time independent perturbation theory?

I am discussing an article

Luigi Garziano, Vincenzo Macrì, Roberto Stassi, Omar Di Stefano, Franco Nori, and Salvatore Savasta, "A Single Photon Can Simultaneously Excite Two or More Atoms", arXiv:1601.00886v2

The article shows that a single microwave photon can excite at once two atoms space-separated, placed in a resonant cavity, the photon energy being equal to the sum of the excitation energies of the atoms.

The initial state of the total system, 2 atoms + 1 photon, is |g, g, 1>, i.e. each atom is in the ground state, and there is 1 photon in the cavity. The final state is |e, e, 0>, i.e. both atoms are excited and the photon was absorbed.

The perturbation theory shows that the system evolution is not unitary - the total energy is not conserved. More exactly, between the initial and the final state the system passes through intermediary states in which VIRTUAL PHOTONS appear, or, in which the initial photon disappears. The calculi in the article show that these states have an impact on the Rabi frequency, s.t. their existence is indirectly testable through the Rabi frequency value.

Bottom line, do we have a rigorous argument against the existence of virtual particles? Can we rigorously deny them?

I have seen the paper: W. Li and J.-X. Chen,

The eigenvalue perturbation bound for arbitrary matrices

, J. Comput.Math. (2006), 141–148 in this regard. Are there any more? Ren-Cang Li (2014). "Matrix Perturbation Theory". In Hogben, Leslie. Handbook of linear algebra (Second ed.) is another survey. Can anybody help me to have more on this topic?

It is known that the exact eigen-functions of sturm-liouville problem are orthogonal with respect to the weight function. But what happens when the solutions are obtained using approximate analytical method?

Do the approximate analytical solutions obtained using asymptotic techniques or using WKB method orthogonal? 

Dear friends, I want study the photo-catalytic effect computationally using Gaussian package... Is it possible...? If it is so, please suggest me something about this. Thanks in advance...

I have even read Cohen Tannoudji book Atom photon interaction but couldn't understand that

I have a system of n atoms and I am using a nonlinear conjugate gradient (Polak–Ribière) to minimize its energy, which is the Lennard Jones potential. I use strong Wolfe conditions for the line search. The code works great and fast for small systems (up to 60 atoms). But for larger systems the method gets very slow due to round off errors. The step size found using strong Wolfe conditions is small and so for large systems each atom must move a very short distance (because this small step is multiplied by atoms positions then added to previous position vector for the whole system) and on a computer the system does not feel it. 

i have just learn about the Wannier function,and i read the paper wannier90: A tool for obtaining maximally-localised Wannier functions.there is a function which refer to the finite difference formulas,after that there is a equation which should use some algebra,but i cant deduct them

just as follow:

I'm just need to clarify some moment. Assume, we have some energy spectrum, and we calculate the correction to it from the usual stationary perturbation theory. The new spectrum is real in common, but corrections to some levels are pure imaginary.

Thus, can we say that these levels becomes metastable under the perturbation influence, and its life-time is determined from the imaginary part of energy? These levels are higher than the ground state, therefore the transition to ground state is possible (and it is not prohibited by any selection rules).

I want to know whether someone know anything about the derivations of KP equation, C-H equation by using reductive perturbation method? or something recommanded to me about the reductive perturbation method?

In QFT we calculate quantities that ultimately depend on the way we normalize spinors; but because in perturbation theories spinor propagators are free then their normalization can be chosen arbitrarily: so in perturbative QFT computations give results that depend on arbitrary normalization. It is custom to choose the spinor square scalar equal to 2m and this gives the correct results; but for instance in condensed state physics the normalization would be to choose it equal to the number of particles, which in QFT would result in choosing it equal to 1 and this would be grossly out of scale. So given that 2m seems correct but not the only possibility, is there a way in which such a normalization could be justified a priori?

 I was trying to do SAPT analysis using MOLPRO program for the H-bonded complex with one monomer triplet and one monomer singlet. There was no error in the output. But I couldn't trust the result because the result was exactly same for triplet and  singlet complexes. Would you guys have any suggestion?Thanks   

Dear Vasp users,

I am working on the electronic properties of 2D heterostructure of TMDC's. For optimization of these heterostructure i have used optB86b-vdw functional. Further in my studies, i want to calculate second order force constants (for phonons dispersion), born effective charges and dielectric tensor for these materials. In order to obtained these quantities i have to employ density functional perturbation theory (DFPT). Can we use vanderwaal functional like optb86b-vdw during DFPT calculations?

Earlier i have tried to use Grimme correction for van der waal interaction (DFT-D2) along with DFPT but i got following error

So, i am bit worried can we use optB86b-vdw along with DFPT?

Please give your suggestions.

With best regards,

Sitansh

In case of time-independent perturbation theory in Quantum mechanics, we find that, the first order correction to the energy is the expectation value of the perturbation in the unperturbed state.

What is the physical implication of this?

can the sort of perturbation theory used in cosmology be used to describe the surface of the earth?

I ran a Opt+Freq calculation on the following NHC Carbene complex using WB97X-D/LanL2DZ/5D 7F in Gaussian09 on a cluster, and I have confirmed a minimum . I then ran a single energy calculation as the following:

The following value is causing the anxiety in the perturbation theory energy analysis of the output:

373. BD*( 1)Fe 41 - C 47     /130. LP*( 4)Fe 41 (564.77 kcal/mol) 0.04 0.290

The Donor is the BD*(1) Fe41-C47 NBO and the acceptor is a LP* (4) NBO on Fe41. According to the main listing of NBOs, NBO 373 is an antibonding  orbital between the Fe41's sp^(1.13)d^(3.87) hybrid orbital and C47's sp^(1.18) hybrid orbital. The coefficients are 0.8524 on Fe41 and -0.5228 on C47.  The LP*(4) NBO is located on the Fe41 as a sp^(0.24)d^(1.31) hybrid orbital.  The interaction between the two orbitals stabilize the complex by 564.77 kcal/mol which is a lot!  I know this is a lot to interpret, but I do not have any software to visualize NBO interactions. If one needs more information, I have attached the .out file. 

In a normal CASSCF calculation using G09 i have the following line in the ouput file

2ND ORD PT ENERGY CV 0.000000 CU -0.085524 UV -0.090730

and i'm not sure to what perturbation it's referring to, the route section in my input reads as follows

#P CASSCF(6,6)/cc-pVTZ

so i'm assuming that the perturbation referred to in the output is a different perturbation as opposed to that of CASPT2.

any information about it, is a great deal of help to me, thanks in advance.

Dear all

In one of my unpublished paper, I used second order perturbation theory to calculate the magnetic anisotropy of a Ni surface. The unperturbated wavefunction was calculated by colinear DFT. The detail formula  can be found in Phys. Rev. Lett. 99, 177207.I think as long as the perturbation converged, the Rashba and Dresselhaus term has been included in the calculation. Because the information of symmetry breaking is contained in the charge density (or orbital occupancy) of colinear DFT. However, somebody told me that I should additionally include a Rashba term to make my calculation suitable for a surface system.  So I am confused. Who is correct?

The question is related to my work on Donor-Acceptor-Hybrides.

Perturbation Theory might help with the answer. The question is related to my work on Donor-Acceptor-Hybrides.

We all know that the Correlation matrix is :

Rxx = E{x.x^H} where E{} denotes expectation and H is the hermitian operator.

In practice, and in most cases, the E{} is replaced by the sample average.

x is an N x 1 column complex vector.

I would like to know how the eigenvalues of Rxx are affected if x were affected by a diagonal matrix C that changes every sample and depends only on a scalar, say 'alfa' i.e.

Rx'x' = E{x'.x'^H} where x' = Cx

or

Rx'x' = 1/N * (C(1)x(1)[C(1)x(1)]^H + ......... + C(N)x(N)[C(N)x(N)]^H )

The quark mixing matrix in the Wolfenstein parameterisation can be viewed as a perturbation of unit matrix with different contributions at different order. Has anyone tried to explain quark mixing matrix as such a perturbation by diagonalising the up and down quark mass matrices using the perturbation theory? Such an approach will start with considering up and down type quark matrices as unit matrices at the leading order and then consider perturbations to it at various orders.

After I read the result (attached) generated by runts GAMES, I found that it didn't run the trans program and coupled cluster calculations.

Does anyone have experience with this? What's the input file of the "tran" program?

Thank you.

Let {$\psi_n$} be a basis in a Hilbert space and $\phi_n$ be a perturbation of $\psi_n$. What is an effective condition for the system of vectors {$\\phi_n$} to be complete? Please, give fresh references (particularly, later than the book T.Kato "Perturbation Theory for Linear Operators")

Some say its a strength of perturbation. Some say its a matter of book-keeping and will anyway have to be substituted equal to 1 at the end. Some books like R Shankar do not even care to introduce g. 

Is anyone familiar with a version of time-dependent perturbation theory in which both the unperturbed Hamiltonian and the perturbation are not Hermitian? The unperturbed Hamiltonian is constant and the perturbation varies sinusoidally with time in the problem we're studying.

What is the method for finding the power spectrum from the distribution of density? Considering I have the distribution of density over a certain volume.

Can null models be applied to comparing time series data? If so, please, offer your methods for doing so.