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QuantumChemistryandSpectroscopy Text book Engel: Quantum Chemistry and Spectroscopy (3 ed., QCS) or Engel and Reid: Physical Chemistry (3 ed.) Chapter8.Quantumchemistry(Engel:QCSchapter15) Let us finally concentrate to actual quantum chemistry. The Hamiltonian is known é Z I Z J e2 h2 2 Z I e2 e2 H = ê- å Ñi - å +å +å 2 m 4 pe | r R | 4 pe r iI j <i I >J 4pe0 | RI - RJ êë i e 0 i I 0 ij ù ú | úû In this we need to know the atomic type and positions. The positions do not need to be exact but they need to be reasonable. The molecular geometry can be optimized later. As said earlier the exact wave functions cannot be solved and thus some approximations are needed. We also utilize the variational principle to find the best trial wave functions from the chosen function class. We already used non-self consistent LCAO trial functions but they are too simple. Hartree-Fock equations The simplest anti-symmetric product function is the Slater determinant Y ( r1 , r2 ,.., rN ) = 1 det | j 1 ( r1 )j 2 ( r2 )..j N ( rN ) | N! Where the φ(r) is an atomic type orbital. We need to write there using some basis functions ξ(r)(simple known functions that contain adjustable parameters. Typically the basis functions are centered to atoms, ξ(r-R). ( ) = ( ) , Now we can insert to the Slater determinant and the basis functions to variation equation (or to Schrödinger equation). After a bit of math we can write the Roothaan-Hall (R-H) equation = Where C contain all the molecular coefficients, F is the Fock matrix and S is the overlap matrix. The Fock matrix is ( =∫ | )= ( ) − = ( ) ħ ( = ∇ − + | ) ∑ ( ) ( ) ( ) | − | = ( ) | − = | ( ) ( ( ) | ) = , , This is quite complex equation since the solution of the R-H equation is hidden to the coefficients J and K. Also the integrals (nm|kl) contain 4 functions (and they are 6 dimensional). The J can be simplified but K not. If there is M basis functions the computations scale as M4. These together are the Hartree-Fock (HF) equations. They cannot be solved directly. One need to make a guess of C(0) and solve K(0), J(0) and F(0) matrixes with this guess. Then the R-H equation can be solved and a new set of coefficients C(1) can be solved. Usually one have to adjust the new C’s a bit but this self-consistent loop usually converges quite well (at least if there is a large HOMOLUMO gap). Solve K(n) J(n) and H Guess C(0) Solve F(n)C’(n+1)=S C’(n+1) Is DC(n)small enough Get new C(n+1)= (1-a)C(n) + aC’(n+1) The next issue is the basis functions, ξ(r). There are several possibilities but the most common functions are gaussians ( )= ( )exp(− ) ( , ) The Pn(r) is some polynomial to make the functions more atomic like and Ylm are the spherical harmonic functions. It would be nice to use atomic type functions (Slater functions) but the J and K integrals become tedious with them. With gaussians these integrals can be done analytically but we need more Gaussians to get good accuracy. It is close to an art to make a good Gaussian basis. Typically we need higher angular momentums than the valence electrons have, e.q. d-orbital for C, O, N etc. We need two (or even more) exponents since in the molecule the wave functions decay from the nucleus is not symmetric. Polarization functions: basis functions with higher angular momentum Double-zeta (DZ), triple-zeta (TZ) functions: basis functions with different exponents Diffuse functions: very broad gaussians, needed for intermolecular interactions. In summary the HF theory and the basis functions will limit the accuracy of the calculations. If the basis is very good and in practice do not cause any practical error to the calculations the results are referred to be at the HF limit. With modern computers and rather small molecules it not difficult to get to the HF limit. But even then the results are not very good since the HF itself is not very accurate. Well some quantities, like bond distances, are very good with HF but for example the binding energies are not. The geometries are good but most bonds are a bit too short. We need to discuss a bit how to optimize the molecules geometry using the HF theory. We can compute the total electronic energy of the system. That depend of the atomic positions. Then we can compute the forces acting on each atom ( ,.. ); =− Once we have the forces several minimization algorithms can be used to minimize the molecules geometry. We can easily find only the nearest minima from the starting geometry. Often the molecule will have several local minima. In simple molecules we can what the minima might be. Unfortunately the number of minima will increase rapidly when the size of the molecule increases. There is not general method to estimate the number of minima of a relatively large molecule but it is very high. For this reason any computational research of proteins is difficult or impossible if we do not have a good guess of the structure. To some extent this is true for solid materials too but still for most molecules we are interested in chemistry this is not a (big) problem. Also the vibrational frequencies are not easy. In fact they are difficult quantities to compute since they are based of very small energy differences. The vibrations can be computed as the second derivative of the total energy = From this matrix we need to find the vibrational normal coordinates and the vibrational eigenvalues. The computations are rather technical but easy to do. The most demanding part is the calculation of the second derivative. In fact any property related to molecules structure can be computed. One very interesting field is chemical reactivity. Reactions equilibrium constant and reaction rate can be computed (but they are not very accurate). The reactants will have a certain lowest energy geometry, the transition state will have an other geometry and the products will again have some geometry. If these geometries can be found the Reactant, Transition state and Product energies can be computed. Unfortunately the relevan energy is not the electron energy but the free energy, F = H – TS, so we need to estimate the entropy. In gas phase molecules and on surfaces this is easy (we need the molecular vibrations) but it is not easy in liquid. In this course we will not go to the details. Beside the structural porperties also the electronic properties can be computed. Of these one very interesting one is the dipole moment. The change of the dipole moment will determine the strength of the vibration adsorption peaks. The HF dipole moments are not very accurate but they are systematically overestimated. Any other electronic quantity can be computed, like polarizations. Maybe surprisingly the electronic energy states and especially the exited states are not very reliable in HF. Summary: HF is the basis of all traditional quantum chemical methods. It is not considered to be very accurate and in any serious quantum chemical calculations some “post HF” should be used. Post Hartree-Fock methods There are many ways to improve the HF method. Most of them are very technical and in this course only the basic ideas behind them are given. But we can first “define” the correlation energy. There is the best non-relativistic total energy with Born-Oppenheimer approximation of the system Etot, the correlation energy is the difference Ecorr = Etot - EHF < 0 This is not very useful definition but if we have different well defined methods to compute the correlation energy, the one which have lowest energy is the best. The conceptually simplest post HF method is the Configuration Interaction (CI) methods. In this method the wave functions is built from several deteminants Ψ( , . . )= Ψ + , Ψ + , Ψ +. .. The new determinants are built form the HF orbitals but the electrons are also placed on exited states. The notation Ψ means that one electron is excited from state i to state a, similarly Ψ means that two electrons are excited from states i and j to states a and b. This wave functions is denoted as CISD (CI with singlet and double excitations). The Ψ is the Slater determinant (or the HF wave functions). All the excited determinants are orthogonal to the HF wf (and each other). In the CI calculations the HF orbitals are kept constant and the a coefficients are optimized. The CISD method is a reasonable solution for the correlation energy for small molecules but for larger molecules it become inefficient. Also for larger molecules CISD is not very accurate and higher terms would be useful but methods like CISDT become computationally very expensive. There will be a huge amount of excitations to be compute. In general the pure CI type methods become too expensive when the molecules size in increasing Scaling Method N4 HF N5 MP2 N6 MP3, CISD, CCSD N7 MP4, CCSD(T) N8 (MP5), CISDT, CCSDT N10 (MP7), CISDTQ, CCSDTQ The CI has also a size consistency problem. This can be illustrated with an example of He dimer. The He atom have only two electrons and thus CISD is an exact method for it. When the He2 is studied at CISD level the theory is not exact since the triple and quadrupole excitation are missing. In general finite-CI accuracy reduce with the size of the molecule and any calculation which deals with molecules association is biased with the size consistency. This is a rather large huge problem for computational binding energies and an unfortunately feature for many Post-HF methods. There are several variations to the CI method, like Coupled Cluster (CC) and multireference (MRCI) methods but they are too complex for this course. Perturbation methods Another approach to correlation it the perturbation methods. The idea is very general and it can be used to several other quantum chemical problems. We can assume that we can solve Hamiltonian H(0) and the full Hamiltonian have a small perturbation V. The perturbation will be scale with parameter l. ( ) = + We can next write the energy and wave functions as a power series of l. = ( ) + ( ) ( ) + +⋯ ( ) = + ( ) ( ) + +⋯ Now we can write hierarchical equation for each power of l ( ) ( ) ( ) + ( ) + ( ) ( ) ( ) ( ) = = ( ) = ( ) ( ) ( ) + ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ) The different level wave functions are orthogonal So we can solve ( ) =〈 ( ) ( )〉 ( ) , Ψ( ), ( ) =〈 ( ) ( ) ( ) etc. ( )〉 ( ) =∑ 〈 ( ) ( ) ( )〉 ( ) ( ) ( ) = . ( ) = 〈 ( ) ( ) − ( )〉 ( ) The perturbation theory can be applied to correlation energy. This is called Moller-Plesset theory (MPn). Then the H (0) is the HF theory and the perturbation is 1 = − − ( − 1 2 ) This clever since it corrects the HF error. The level (2) correction is ( ) = , [( | ) − ( | )] + − − The i,j denote occupied state and a,b unoccupied states. This the first real correction and it is called the MP2 theory. (The E(0) + E(1) is in fact the HF energy). The perturbation theory can be to arbitrary level. The different levels are denotes as MPn. Of these the most important is the MP2. It is rather fast and causes a significant correction to HF. The MP3 is a disappointment since it does not corrects much the MP2. The MP4 is somewhat better but it is already very expensive. The higher terms are not used and in some model systems it has been shown that the MPn series do not converge. Computationally the MP2 with small approximations can be implemented very effectively. The RI-MP2 method is much faster than and essentially as accurate as true MP2. All practical MP2 calculations for larger systems are done with RI-MP2 (or its variants). In big RI-MP2 calculations the time consuming part is the HF. The perturbation method is very general and it can used in several quantum chemical problems. One application is the CI (and CC) methods where the higher order excitations can be solved with perturbation theory. The CCSD(T) method is the “golden standard” of quantum chemistry and it has the same scaling behavior as MP4. Last the basis set needed for well converged CI and MPn methods is larger than in the HF calculations. This is bad news since these post-HF methods have poorer scaling behavior than HF so the computations become quickly very time consuming. Density Functional Theory The traditional Quantum Chemical methods will approach systematically the solution of the correlation energy but they lead to very heavy computational procedures. The Density Functional Theory will utilize another approach. In its hart is a proof that the ground state of an electronic system depend only on the electron density. Y ( r1 , r2 ,.., rN ) a Y [n ( r ) ] This wave function is much much simpler than the true wf. The problem is that we do not know the equations to solve the wf. The basic ideas are form Kohn, Hohenberg and Sham (1963, 1964, Kohn Nobel prize 1998). Even we do not know the correct equation it is useful to try to find a good approximation for it. Kohn and Sham showed that a single Slater determinant type wf with simple model for the Exchange and Correlation (XC) energy will give good results. The formulation is quite similar as for the HF. =∫ = ( ) − = ħ = ∇ − = + ∑ | ( ( ), ∇ ( ) , . . ) ( ) ( ) | − | − ( ) | ; ( ) = = , , ( ) ( ) The difficult part is the XC energy. It is not known but several approximation are done. The simplest Local Density Approximation (LDA) depend only on density, EXC,LDA[n(r)]. This functional can be obtained from exact calculations of homogenous electron gas. The LDA work reasonably well and it is better than HF (with few exceptions) but electrons are NOT homogenously spread in molecules. The next level is the Generalized Gradient Approximation, EXC,GGA[n(r),∇ ( )]. The GGA models are much better but there is no simple (or even non-simple) procedure to build the GGA functionals. For this reason there are a huge amount of different GGA models. The most common are BLYP and PBE. Even there are far too many GGA models, the DFT-GGA method have one very strong advantage. It scales as N3. It is much more accurate than HF but it is much faster. With DFT we cannot get systematic improvement of the accuracy but it can be applied to big systems are the results are almost always good (same level as MP2). One can also mix the HF and DFT theories. These are called hybrid methods and they also are rather accurate. The most common of them is the B3LYP model also PBE0 model is good. As DFT cannot be improved systematically the results matters. The DFT wf’s are variational but the models are not, so we cannot judge the quality of results with the correlation energy. The results can be compared to experiments or to more accurate quantum chemical methods. Below there are few tables. They show that B3LYP method is usually as good as MP2. Even the tables shows only one DFT method the general trend is correct. In most cases the DFT methods are as good as or better than MP2. There are more tables in Engels book. Almost all methods the bond distances are quite good. For energies the HF or LDA are not reliable whereas most GGA’s and correlated quantum chemistry methods are good. The reaction barriers are challenging form computational methods and in them the GGA methods work rather well. Naturally the GGA methods have some problems. One of them is the van der Waals interactions (or dispersion) which is important in weakly bonded systems. Basically HF and GGA will do not have them and they have to be added using additional (empirical) terms. In most cases such PBE-D3 models works well. DFT have also problems to describe the hydrogen correctly. Like HF the DFT models do not give good HOMO-LUMO gap. There is a lot of research focusing on development of the computational methods. In traditional quantum chemistry one area is on development fast computational algorithms for the known methods. In DFT the development is rather slow. Now there is quite a bit of interest of beyond-DFT methods. Like RPA, MP2 etc. On the other hand both these methods have enormous amount of applications covering almost all fields of chemistry. DFT can be used to model single molecules, clusters of atoms, surfaces, solid systems and liquids. Systems up to 100.000 valence electrons can be computed. An emerging research field is quantum mechanical materials screening where properties of materials (or molecules) will be evaluated computationally and this information is used to help the synthetic work.