Recently I use the siesta to calculate the optical properties of doped graphene nano-ribbons. But I want to know whether the results are correctly or not. as the the Marty explain if the DFT eigenenergies are right (usually fixed by placing a scissor operator). Now the question is how to determine the optical gap of doped graphene nano-ribbons ?
在2011-03-05 04:12:30，"Marty Blaber" <[email protected]> 写道： Hi All, This email got a little longer than I expected. It is a general description of how you get to the equations that Siesta uses to calculate the optical properties, what they mean, and what the problems are with such a method. I tried to make it accessible to everyone. Please comment with questions/suggestions/corrections. Siesta uses first order perturbation theory to calculate the optical properties of materials/molecules. In essence, the idea is that you solve the problem of a photon perturbing a Model, single particle Hamiltonian, and then use DFT energies and wavefunctions to plug into the resulting formula. To get the formula, you start with a model ground state hamiltonian H0, and add a perturbing potential V: H0 = p^2 + U(r) = Ground State H= H0 + V(r) = Ground + Perturbation p is the momentum operator, U is the crystal potential, V= Vext + Vs where Vext is the time dependent perturbation, and Vs is the screening interaction due to the perturbation. So you solve the Liouville equation for the above system (by splitting the density matrix into a ground state and perturbed part), and after some algebraic and fourier hand waving you get something like [1-7]: epsilon = Integral over all k of { [ ( Difference in occupation Numbers) / ( Difference in eigenvalues - impinging photon frequency ) ] * < psi_initial*(k) | p | psi_final(k) > < psi_final*(k) | p | psi_initial(k) > }. The epsilon above is the solution within first order time dependent perturbation theory (the first order part comes from approximating the commutator of H with the perturbed density matrix in the Liouville equation). However, there is no exchange or correlation in the solution!!! You can sort of correct the lack of exchange and correlation by replacing the eigenvalues and psi's (which are solutions to the single particle Hamiltonian above) with ground state DFT eigenvalues and wavefunctions. Now at least the energies and the character (s,p,d,f) of the eigenfunctions are much better, so you get a very reasonable answer in many situations. But you must rely heavily on the correctness of the eigenvalues and eigenfunctions from ground state DFT. (And there is still no exchange or correlation in the response function itself.) The bits in the Bra's and Ket's are the momentum matrix elements or transition dipole matrix elements. In the end the optical response boils down to the difference in the s,p,d,f character between the initial and final states. i.e. pure s to pure p has higher probability than pure s to little bit of p with lots of d. These transitions are weighted by the occupation probability, and there is a resonance when the optical frequency is the same as the difference in energy between the bands/states. I should point out, that this is essentially the same as the RPA or random phase approximation, and is also called DFT-LDA optical response or the Sum over states approach. If you want the details of the derivation read [1-7]. (Above I only discuss interband transitions, Siesta also includes intraband transitions and it gives these to you in the form of the plasma frequency in EPSIMG for metallic things only) So, now the problems with this method. First, the DFT eigenenergies are not always right. For semiconductors this can be fixed by placing a scissor operator in there i.e. rigid shift of the bands so that the gap is correct. For metals, particularly transitions metals, the ordering of the bands is sometimes wrong for some momenta and the dispersion is not quite right. So people use this "lambda" parameter, which just stretches or compresses the bands over some frequency range to get the dispersion correct. The method is inherently single particle/ single reference/ single determinant as mentioned by N H and Gregorio. So most of the excited state properties are not correct. The character of the HOMO and LUMO has been shown to be rigorous, but beyond that... it depends on the system in question. Importantly for semi-conducting CNTs, there are no excitons! The response function has no exchange or correlation, therefore, you cannot create a bound electron-hole pair! This is a very important aspect for some types of CNT's. See for example [8]. I quote : However, measured optical transition frequencies deviate substantially from theoretical predictions based on one-particle interband theories. This deviation is not unexpected since many-body interactions should play a vital role in reduced dimensions. Answers to questions: caqhero I used the siesta to calculate the optical properties of CNTs. The reviewer ask the method I used. and he say that the Ground-state DFT can not accurately describe optical properties. What you did is used first order perturbation theory to calculate the optical response using ground state dft eigenvalues and eigenfunctions. It depends on exactly what aspects of the response you are interested in as to whether you can say they are reasonable. If you are looking at trends, and you know you're system is not excitonic, then you may be fine!! caqhero In that paper, it is said that the first-order time-dependent perturbation theory to calculate the dipolar transition matrix elements between occupied and unoccupied single-electron eigenstates, as implemented in SIESTA, but I can not find it. in the manual, it is said that The optical calculation is performed using the approach based on the dipolar transistion matrix elements between different eigenfunctions of the self-consistent Hamiltonian. Both of these statements are the same. The theory is first order time dependent perturbation theory, but you use self consistent ground state DFT energies and eigenfunctions to plug into the transition dipole matrix elements. TDDFT may help, I think it depends on whether or not you are using periodic boundary conditions or not. For a molecule or isolated tube it should in theory give exact energy differences, but whether it captures excitonic effects I don't know. This paper seems to say yes if you do some juggling: http://pubs.acs.org/doi/pdf/10.1021/ct100508y If periodic boundary conditions are in place, I think you are limited to using GW or BSE (see below). I hope this helps, sorry it was so long. There are references below. Claudia Ambrosch-Draxl's papers are pretty good. There are some great slides here about DFT, GW (for correcting DFT eigenvalues) the Bethe Salpeter Equation for studying excitons: http://www.tddft.org/TDDFT2010/school/gatti12.pdf and one about CNT's and excitons http://www.slidefinder.net/F/First_principles_prediction_electronic_excitations/17212507 Cheers, Marty [1] H. Ehrenreich and M. H. Cohen, Physical Review 115, 786 (1959). [2] N. Wiser, Physical Review 129, 62 (1963). [3] S. L. Adler, Physical Review 126, 413 (1962). [4] P. Nozires and D. Pines, Physical Review 109, 741 (1958). [5] P. Nozires and D. Pines, Physical Review 109, 762 (1958). [6] P. Nozires and D. Pines, Il Nuovo Cimento (1955-1965) 9, 470 (1958). [7] C. Ambrosch-Draxl and J. O. Sofo, Computer Physics Communications 175, 1 (2006). ALSO http://arxiv.org/abs/cond-mat/0402523 [8] C. D. Spataru et al, Physical Review Letters 92, 077402 (2004). [9] R. M. Martin, Electronic Structure: Basic Theory and Practicle Methods (Cambridge University Press, 2004). 2011/3/4 Gregorio García Moreno<[email protected]> NH Respect to TD implemented in SIESTA, (perhaps I am wrong) by in a congress someone commented me that Miguel Pruneda (Pablo Ordejon's group) has a private version with TD implemented in SIESTA. From references before mentioned, it can be seen that TD along with long range corrected (LR) functional may be a good selection to obtain accurate theoretical values. However, I have tested LR corrected funcional implemented in Gaussian for same organic compounds and they don't give accurate orbital energies. However, they yields accurated optical bandgaps calculated as HOMO- LUMO transition energy using TD-LR. El 3/4/2011 12:10 PM, N H escribió: At first thanks for the references Gregorio!! :) Some of them I really dont know until now! The question I alwaus rise on this issue is that, sure, the electronic density derived from the Kohn-Sham orbitals do have the information about all electronic states for a given system in a given external potential. But, as most refereces you already cited above, to extract this information it is necessary to go beyond the time-independet approach. And that is were the discussion comes back to the caqhero question. There might be particular TD siesta implementations ... but they are particular! Concerning on the relation between IPs/EAs and HOMO/LUMO ... sure, you are absolutely right. But this LUMO just assumes a physical meaning if you are putting one electron there and calculating the EA. Otherwise, this orbital will never be taken into account while minimizing the electronic energy. Caqhero To be brief ... the official SIESTA package is not an implementation of Time-Dependent DFT. :( Cheers NH 2011/3/4 Gregorio García Moreno<[email protected]> There is some controversia about the physics meaning of Kohn-Sham ortibals, i.e. Koopman's theorem is no sastified. However,Perdew, using Janak’s theorem has proven a connection between IPs / EAs and HOMO / LUMO energies, respectively P. Perdew, in: R.M. Dreizler, J. Providenca (Eds.), Density FunctionalMethods in Physics, Plenum Press, New York and London, 1985, E. Jansson, P. C. Jha, H. Ågren, Chem. Phys. 330, 166, 2006.). Furthermore, it is known that long-range corrected functionals give accurate orbital energies and satisfy Koopman theorem. Tsuneda, T.; Song, J. W.; Suzuki, S.; Hirao, K. J. Chem.Phys., 2010, 133, 174101. Rienstra-Kiracofe, J. C.; Tschumper, G. S.; Schaefer, H. F.; Nandi, S.; Ellison, G. B. Chem. Rev. 2002, 102, 231-282. Jacquemin, D.; Adamo, Carlo, J. Chem. Theory Comput, 2011, 7, 369-376. JacJCTC08. D. Jacquemin, E. A. Perpète, G. E. Scuseria, I. Ciofini, C. Adamo, J. Chem. Theory Comput., 2008, 4, 123.WonJCTC10. b. Wong, T. H. Hsieh, J. Chem. Theory. Comput, 2010, 6, 3704.JacJCP07. D. Jacquemin, E. A. Perpète, G. Scalmani, M. J. Frisch, R. Kobayashi, C. Adamo, J. Chem. Phys., 2007, 126, 144101. Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, “A long-range-corrected time-dependent density functional theory,”J. Chem. Phys.,120 (2004) 8425. Sorry, perhaps we are moving away from the Caqhero's problem El 3/4/2011 11:08 AM, N H escribió: Well ... for sure there are lots of studies using DFT to calculate such properties. Another question is whether this results are meaningful or not... and that is controversial. The problem is that some people don´t see much meaning on conduction bands calculated with single reference methods, since their eigenvalues are not really taken into account while diagonalizing the system's Hamiltonian. Another issue is the meaning of the Kohn-Sham orbitals themselves. It has been demonstrated - at least for molecules - that they do have the same simmetry properties as HF orbitals and that the also agree with the Koopman's theorem after rescalling. I also know that there are some recent works (see the link) on a DFT version of the Koopman's theorem, but in order to verify that you have to go beyond standard DFT: http://www.ingentaconnect.com/content/nrc/cjc/2009/00000087/00000010/art00014 Given that, you can calculate the optical properties of any material with DFT if you think that the conduction bands are meaningful ... but you are going to have a hard time if your referee think it is not true. Cheers NH 2011/3/4 Gregorio García Moreno<[email protected]> Hi I have never used SIESTA to calculate optical properties, but recently have accepted a propious work where electronic structure of conducting polymers: band diagrams, bandgap, bandwiths, effective mass, ect. All these properties are related with optical properties, and the reviewers didn't say anything about the methodology (DFT and SIESTA) I can't help you, but you can look for other works where DFT implemented in siesta is used to assess optical properties. Besides, respect to DFT there are a lot of works which use DFT to calculate optical properties of photovoltaic cells, oleds, synthetized dyes, etc. I think that that all works of nowadays on optical properties use DFT. I know that Carlo Adamo's group are working in the development of new fuctionals to assess optical properties. Unfortunately, all these functional are not implemented in SIESTA. On the other hand, there are a lot of scale factors in bibliography to correct theoretical values in concordance with experimental ones. Sorry, I can't help more Gregorio El 3/4/2011 9:16 AM, caqhero escribió: Recently, I used the siesta to calculate the optical properties of CNTs. The reviewer ask the method I used. and he say that the Ground-state DFT can not accurately describe optical properties. I confuse the method for optical calculation in siesta. can any body tell me the details ? how can I reply the reviewer ? is it reliable using the siesta to calculate the optical properties ? any suggestion is appreciated ! thank you ! -- Gregorio García Moreno PhD student Department of Physical and Analytical Chemistry University of Jaén
