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# Perturbation Theory - Science topic

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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.
Something’s obviously missing here. The terms that describe the contribution of opposite spins" can come from, either the sector of total spin-1, or the sector of total spin-0. The terms that describe the contribution of same spins" can only come from the sector of total spin-1. So the statement that the second order correction can be divided in the way described is incomplete, because it doesn’t take into account the total spin. Two spin-1/2 combine to a spin-1 sector and a spin-0 sector.
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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+...+An for a fixed natural n? Many thanks in advance. I am looking forward to your suggestions and recommendations on this topic.
Hi professors. I hope you are doing well. This is a mathematical article that proves a new recurrence relation that is fundamental for mathematics. The article proves also that four infinite series are equivalent. Hence, this article opens new opportunities to demonstrate and develop new mathematical findings and observations. This is the link: https://www.researchgate.net/publication/364651911_A_useful_new_equation_of_four_infinite_series_and_sums_by_using_a_new_demonstrated_recurrence_relation
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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?
Helmhotz perturbation -
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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).
In the first half of the 20th century some theorists (e.g. Heisenberg, Brouwer, etc.) tried to develop the concept that space itself has a metric (the minimal length scale of discrete space). The size of the metric was thought to be ≈ 1 x 10-15 m because of the size of the minimal wave length of electromagnetic waves and the diameter of particles. However, 1 x 10-15 m is too large in relation to both amplitudes of 1 electromagnetic wave so the minimal length scale must be ≈ 0,5 x 10-15 m or a bit smaller. The consequence is that we cannot detect phenomena smaller than ≈ 0,5 x 10-15 m.
We are aware of the existence of discrete space because without a spatial structure there are no observable differences in the universe. But observable reality is created by discrete space and we know it because of the spatial differentiation of force fields (the general concept of QFT). The consequence is that the nature of everything we can observe and detect are mutual relations (“proved” by the formalism of QM). Thus we don’t measure the bare existence of the spatial units of discrete space, we measure the mutual interactions between the units, the exchange of variable properties.
It is obvious that these mutual interactions of the units of discrete space cannot “split” a unit of discrete space (even the ancient Greek philosophers reasoned some 2500 years ago that there is a limitation on reductionism). The magic word in theoretical physics to solve the problem is “asymptotic freedom”. We really like it to give problems a fuzzy name so we can keep our ambiguous concepts.
Recently astronomers have observed large regions of gravitational polarization in the early universe (Cosmic Microwave Background radiation by the BICEP2 Collaboration). That is a problem because the CMB radiation is the exchange of electromagnetic waves between the Hydrogen atoms in the early universe. So how is it possible that there are already regions of huge gravitational fields if there are no stars, etc.? There are also observations of “full grown” galaxies that already existed about 0,7 billions years after the proposed “big-bang”. These galaxies have an enormous black hole in the centre so cosmologists have termed these black holes “primordial” black holes. But thanks to the BICEP2 measurements we now know that our universe created the enormous black holes first, before there was the creation of Hydrogen atoms.
Now there is your question about the existence of quarks. Are they real or are they the result of tricky theoretical constructions with the help of “asymptotic freedom”? Moreover, can QCD elucidate why vacuum space created enormous black holes before the Hydrogen atoms emerged from vacuum space around the black holes? I will read your paper about ontology and physics. ;-)
With kind regards, Sydney
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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
@Stephen Mason
In fact I mean
y"(x)= L x^2
In a simplified language second derivative of
y with respect to x = L x^2
Where L = constant
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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.
A standard trick in these cases is to use the Sherman Morrison formula
However, the inverse of a sparse matrix does not have to be sparse and in particular one does not want to store inverses for sparse large matrices, so that the formula would rather have to be applied to the action A^-1b of the inverse on the right hand side of the linear system, so as to correct the solution to the original linear system A^-1b with a hopefully limited amount of operations.
Please notice that this is a very generic comment, I am sure somebody in the sparse solver community will have studied the problem in much greater depth.
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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?
For heavy metal, sometimes force theorem fail to describe MAE in heavy metal.
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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.
Dear Asheesh,
In my opinion an analysis of the output file does not show any significant interaction (from unit1 to unit2 and vice versa) between the two fragments, that indeed are quite far (> 3 Å).
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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?
Using NBO to try to characterize H-bond interaction is not a good idea. The main nature of typical H-bond is electrostatic, however this vital component cannot be faithfully revealed by E(2) analysis. In addition, even the charge-transfer component in H-bond can also not be correctly reflected by E(2) energy, as demonstrated in J. Phys. Chem. A, 2017, 121 (7), pp 1531–1534.
There are lots of much better ways of demonstrating existence and revealing strength of H-bond, such as Atoms-in-molecules (AIM), noncovalent interaction (NCI), Independent Gradient Model (IGM), electrostatic potential on van der Waals surface and so on. They are fully supported by my wavefunction analysis code Multiwfn (http://sobereva.com/multiwfn). If you are a Gaussian user, the .fch or .wfn/wfx file can be directly used as input file. Please check Section 4.A.5 of Multiwfn manual for a brief review of all methods supported by Multiwfn that may be used to analyze weak interactions.
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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.
Consider A=(1,0;1,0)A=(1,0;1,0) and b=(1;1)b=(1;1). Let xx be any solution of this, so x=(1;t)x=(1;t) for some tt.
Now, perturbe this to (1,0;1−1/n,1/n2)(1,0;1−1/n,1/n2) and bb unchanged, so pertruebed by 00 (but one could also impose some nontrivial perturbation). The perturbation(s) tends to zero. But the (only) solution is (1;n)(1;n). Thus xnxn does not converge, and one seees that the xnxn and the inverse (in this case it is actually invertible, but one could expand the system to avoid this) of the perturbed system are not bounded.
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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.
It doesn't matter, and the E(2) is meaningful. If you understand how the algorithm for generating NBO orbitals works, you will know that these two NBOs in fact are not yielded simultaneously, and therefore their relative phase is completely arbitrary.
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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.
I have attached a pdf file with this reply. Where I have written a pedagogical review of the second order time independent degenerate perturbation theory. If anybody have any question or comment please free to ask/suggest to me.
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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.
Practically the argument follows the extinction characteristics. I think the in silico treatment also will follow the same.
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I am looking for equations to do this calculation and for a book to cite it in my upcoming papers.
Many-Body Methods in Chemistry and Physics MBPT and Coupled-Cluster Theory
Part of Cambridge Molecular Science
• Authors:
• Isaiah Shavitt, University of Illinois, Urbana-Champaign
Rodney J. Bartlett, University of Florida
Many-Body Theory Exposed!
Propagator Description of Quantum Mechanics in Many-Body Systems Willem H Dickhoff (Washington University in St Louis, USA), Dimitri Van Neck (Ghent University, Belgium)
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Non-Linear waves propagating in plasmas
It implies the shape conservation property of the propagating KdV soliton. In other words, the plasma perturbations propagate in such a way that the strength of the nonlinear wave steepening is always proportional to the linear dispersive wave broadening on the spatiotemporal scales of our observation.
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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?
Is there any real life examples to discuss such cases?
Perturbation theory can refer to many different techniques in different fields, for instance differential equations, hamiltonian mechanics, operator theory... Generically you may have situations in which you have a good knowledge of the system, or a situation in which your system is easy to describe and you want to know the behaviour of the system in a more difficult situation by spliting your system in your known part plus a perturbation that may or may not be time dependent. For instance in Hamiltonian mechanics you may have a reference hamiltonian that is integrable and you want to know what happens when you perturb the system, you may ad a time dependent or a time independent perturbing Hamiltonian. It does not mean that your system won't depend on time, it's is just that the Hamiltonian that you are adding doesn't depend on time.
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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?
Dear Kassner,
I think that you are too strict, unpolite and no totally right in your comments on Sofia. Let me summarize some points:
1. This paper is purely theoretical sent the 26 of January and published the 22 of July. Almost half a year under referees that we suppose real knowledge on the subject. In fact they were quite right finishing the paper by the sentence: "might be effective in producing the simultaneous excitation of two or three referees with a single manuscript". Congratulations!
2. This paper was chosen in the physics focus and the authors suggest a quite straightforward experimental procedure to follow their model.
3. Thus it is, at least for me, quite strange that experimental results were not claimed or published, in what that I know confirming it.
4. The conservation of the energy that you say that the authors refer is not so clear for me and obviously not followed in the middle of the physical process where a virtual photon partner of the real one is needed.
5. In any case, I think that this is not enough for saying to Sofia:
Your insistence on non-unitarity plus your unnecessary urge to contradict Stefano who made an entirely correct statement give away your lack of understanding. Nobody who understands the matter would have done this.
6. At least a deeper scientific proof will be necessary and no just some words as you have said involving another person as Stefano, who could respond directly to Sofia.
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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?
Respected Sir,
Kindly go through the following book- Jorge Nocedal, S. Wright. Numerical Optimization. Springer-Verlag New York, 2006. ISBN: 978-0-387-30303-1. 2nd edition.
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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?
Logically, If the solutions are the same, orthogonality property would still be  valid irrespective of the methods that yielded the solution. However, you can subject each of the equivalent solutions to orthogonality test to be certain.
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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...
Thank you very much... for your kind valuable replies.
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I have even read Cohen Tannoudji book Atom photon interaction but couldn't understand that
Thanks Osamah Nawfal Oudah and Riccardo Guareschi for your help. But the problem is that:
Let us suppose H(J) is the actual Hamiltonian and where J is the parameter. When J at some value say Jo has degenerate Eigen values of the Hamiltonian. If I introduce some perturbation to the system then how the Jo values shift.  How this shift is caught by the level shift operator? Or there is any other way to catch this shift?
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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.
Dear Hamza,
-Efficient BackProp - UCSD CSE
-SOLVING LINEAR ALGEBRAIC EQUATIONS CAN BE ... - Project Euclid
-On the robustness of conjugate-gradient methods and quasi-Newton ...
Best regards
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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:
Dear Cheedge,
I think the software you're looking for is Wannier90 (http://wannier.org/), which is a particular program for producing maximally-localised Wannier functions from a given set of Kohn-Sham states (and some starting projections onto localised states). It is available freely under a GNU Public License. Nicola Marzari (lead author of the paper's Behnam referred to) is one of the Wannier90 developers, and if you look at:
You will find many other papers explaining the method and highlighting applications of it to problems in condensed matter physics.
All the best,
Phil
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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 checked that for the infinite well of width Pi. One has to be careful how to evaluate
cos(Pi n) cos(Pi m) cos(Pi p)
for integer m,n,p. If one is not careful, symbolic algebra can indeed produce imaginary <m|V|n> where V=sin(p x) and one of m,p,n is the sum of the other two, e.g. for the cases p=m+n or for p=m and n=2*m. (In these cases, numerator and denominator  in a quotient assume zero values. Replacing cos(Pi p) by exp(I*Pi*p) can trigger the imaginary results.) The correct result however for all such problematic cases is real. This may be seen by perturbing the integer value of p by a small parameter epsilon and  doing series expansions in this parameter. See the attached sheet.
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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?
I don't see exactly what do you mean by "reductive perturbation method" but you can find a rigorous derivation of KP and CH equations (among others!) in the excellent Book by David Lannes on Water Waves, published by the AMS.
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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?
oh, nice, thanks... on the other hand, how can the results be independent on the choice? The Lagrangian density is proportional to the spinor squared and to get the results we integrate over the volume but not divide by the spinor squared: so the spinor squared must still be present and that is normalization-dependent, right? Can you elaborate on this, please?
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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
It is not so that "the results are not reliable" - they just cannot be produced at all. However, there are people how work on open-shell SAPT, so this option might be available in Molpro in the future
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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
ERROR: approximate vdW methods are not implemented for DF-PT.
So, i am bit worried can we use optB86b-vdw along with DFPT?
With best regards,
Sitansh
Hi Prashun,
Thanks for your reply. I will try to run DFPT with optB86 for test system and see if we get some error or not.
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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?
Dear Basabendu,
Glad to be of assistance. If you would include corrections to all orders in the time-independent perturbation, you would obtain (exactly) the new stationary energy eigenstates of the full, perturbed Hamiltonian. In that sense, there is no reference to time dependence in the theory itself - after all, it's a time-independent theory. But for a fully adiabatic perturbation, you could also apply it to track the evolution of a given state vector, since knowing the instantaneous eigenpairs would be enough. (Having said that, my earlier remark on the quench may seem a bit contradictory, but it was just a suggestion where the lowest-order energy correction could have a concrete sense, albeit only very momentarily.)
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can the sort of perturbation theory used in cosmology be used to describe the surface of the earth?
The question sounds rather vague. Nevertheless, I'll try to present some thoughts.
1) In cosmology we use general relativity (GR). GR deals with internal geometry of space-time. Internal geometry is 4-dimensional geometry itself. It is not geometry of some surface in space of higher dimension.
2) The earth surface geometry is external geometry. It is geometry of 2-dimensional surface in 3-dimensional space. E.g. external geometry is able to distinguish a mountain and a ravine. From the point of view of internal geometry a mountain and a ravine are just the same.
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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:
#p rwb97xd/lanl2dz/auto pop=(nbo,savenbo,full) geom=connectivity formcheck gfinput
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.
This calculated energy  (564.77 kcal/mol) depends on the basis basis set . I would not  overestimate its meaning. It just tells you that the little amount of charge in
373. (0.13603) BD*( 1)Fe 41 - C 47 is polarized towards Fe
but I would not  derive any conclusion about  this numerical value.
I send you the test performed using 6-31G* basis set for all the atoms (Fe included).
The large second order energy disappears.
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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.
As Oleg wrote:
CV referred to core-valence of the electron orbital subspace of electron correlation, CU - core-unoccupied subspace, UV - unoccupied - valence subspace.
The following is part of a normal output file of CASSCF calculations.
-------------
2ND ORD PT ENERGY CV 0.000000 CU 0.000000 UV -0.000186
TOTAL 0.499662
ITN= 1 MaxIt=999 E= 0.4998474810 DE= 5.00D-01 Acc= 1.00D-05
ITN= 2 MaxIt=999 E= 0.4660556090 DE=-3.38D-02 Acc= 1.00D-05
ITN= 3 MaxIt=999 E= 0.4999700399 DE= 3.39D-02 Acc= 1.00D-05
ITN= 4 MaxIt=999 E= 0.4996883687 DE=-2.82D-04 Acc= 1.00D-05
ITN= 5 MaxIt=999 E= 0.4996964720 DE= 8.10D-06 Acc= 1.00D-05
... DO AN EXTRA-ITERATION FOR FINAL PRINTING
FINAL EIGENVALUES AND EIGENVECTORS
VECTOR EIGENVALUES CORRESPONDING EIGENVECTOR
1 -1.3618045 -0.43997634E-02 0.99633020
-0.85479675E-01
MCSCF converged.
-----------------------
Rafik
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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?
Hi!
We know that, the electric field of core is seen as magnetic filed in rest frame of electron as   B= - Gama ( beta x E), where beta=V/C. It is internal SOC and is written as:
HI= - meyouB. ( B.S) where S is spin.
When electron jumps between two neighbor atoms, and there is an external electric field, Rashba term appears as:
HR= - lambdaR . (sigma . (E x p)) , where sigma is Pauli's matrices and p is momentum.
When one apply an approximation on Dirac equation, she/he can show that these terms will be appeared in Hamiltonian.
Now, if your DFT code, uses complete Hamiltonian i.e.,
H=H0 + HI + HR
and assumes HI and HR are perturbation terms it is not necessary you consider them again. But if not, you should add them.
Best Regards
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The question is related to my work on Donor-Acceptor-Hybrides.
So far I found an article in the "Bunsenmagazin 4. Jahrgang 4/2002. They cite: M. Kasha, H. Rawls and M. A. El-Bayoumi "The exciton model in Molecular Spectroscopy," Pure applied Chemistry, 1965, p. 371-392; S. Kirstein, H. Moehwald, "Exciton band structures in 2D Aggregates of Cyanine Dyes", Advanced Materials, 1995,7, p. 460-463. I fear that this is not ongoing enough.
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Perturbation Theory might help with the answer. The question is related to my work on Donor-Acceptor-Hybrides.
In addition to excitonic interactions, charge transfer also gives rise to non-radiative decay pathways in aggregates that are not found in the individual chromophores. Charge transfer is important for explaining why the excited-state lifetimes of aromatic excimers are generally shorter (10s - 100s of ps) than the monomer lifetimes (~10 ns). However, excited-state lifetimes of aggregates are not always shorter than those of the constituent monomers! Stacked nucleobases like the ones found in single-stranded DNA (a kind of aggregate) form charge transfer excited states that have much longer lifetimes (10s - 100s of ps) than single bases (~100s of fs).
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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 )
I thought that C is a diagonal matrix with non-random diagonal. If that is not the case, then, if C is independent of X (which i think is the case for your problem since diagonals of C are functions of alpha and alpha seems to be independent of X), the correlation matrix will be E(CXX^H C^H)=E(E(CXX^HC^H|C))=E(CR_xx C^H)=E(T) where the (i,j)th element of T is t_{ij}=C_ii.C*_jj. r_xx(i,j). So E(t_ij)=E(C_ii C*_jj)r_xx(i,j). From your given information, E(C_ii C_jj*)=E(exp(-2*pi*i(i-j)alpha)) which I think you can find easily.
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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.
Lattice simulations of 5D theories, for simulating 4D chiral fermions are possible and have been done-but it depends what one is after. For the phase diagram they're not too difficult; for studying global properties, relevant for QCD they're much harder: cf. here
for a review of one topic.
For condensed matter applications, studies of 3D theories relevant for 2D chiral fermions should be much more feasible.
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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.
At line 141 of file tran/main.F (unit = 2, file = 'vectb.data')
Fortran runtime error: End of file
open 11
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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")
As a continuation of my previous post: yet another condition:
C) 1- D is a  compact operator and D is injective. This is a less restrictive but maybe harder to check version of B.
(proof: 1  -  (1-D)  is a Fredholm operator with index = index(1)  = 0, so just as in the finite dimensional case, injective implies surjective and therefore invertibility.)
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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.
Perhaps you meant to write H = H0 + g H1?  I'll assume that's what you meant in my response.
Typically, H0 is a Hamiltonian whose eigenvalues and eigenvectors are known a priori and one would like to use this information to find the eigenvalues and eigenvectors of H.  Using perturbation theory, one can find the eigenvalues and eigenvectors of H as a power series in g.  Then, one has to study the convergence of the power series to determine if the series is convergent for any particular value of g, including g=1.  It is not a given that any perturbative solution will be valid at g=1.  In fact, it can happen that perturbative solutions have zero radius of convergence and are only valid for g=0.
As for whether you put the factor of g out in front or not really makes no difference.   If you don't like to pull out a factor of g, rescale all the eigenvalues of H1 appropriately.  Regardless, there will still be a radius of convergence for the perturbative solution of H and if there is no explicit g in the problem then it can be expressed in terms of the ratio of the largest eigenvalue of H1 to the largest eigenvalue of H0 and as that ratio tends to zero, then the solution of H tends to the solution of H0.
I think this wikipedia article is reasonably will written and might be useful for you
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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.