Methods for Quantum Simulation - Part 1

hi i'm yukio kawashima i'm a quantum computing application researcher at ibm tokyo research lab today i'm going to talk about the methods of quantum chemistry simulations in the introduction i'm going to show you what quantum chemistry simulation can do and then i'm going to talk about the schrodinger equation and for an oppenheimer approximation to show you what kind of equation we want to solve in quantum chemistry simulations and next i'm going to talk about the quantum chemistry methods the quantum chemistry methods that i'm going to talk about are shown on your hand side i'm going

to talk about the hartree fog method semi-empirical methods correlation methods which include perturbation theory configuration interaction couple cluster methods and density functional theory i am going to also talk about the basis that we use in quantum chemistry simulation and atomic ortho integrals we compute in the quantum chemistry solution to understand all this in just two hours is extremely difficult so therefore the goal for me today is to give you insight of what quantum chemistry simulations are doing by the end of this talk i hope you'll be more interested in quantum chemistry simulations and hopefully you'll

you want you'll be willing to run quantum chemistry simulations on your own now i'd like to start the introduction what can we do with quantum chemistry simulations the first example is tackling the chemical reaction so the the chemical reaction goes from left to right and the vertical axis is showing the energy so the chemical reaction starts from this reactant forming aldehyde molecule and it goes through the transition state and end at the product so the product here is the hydrogen molecule and the carbon monoxide molecule by using quantum chemistry simulation we can calculate the energy

of the reactant transition state and the product by doing so we can calculate energy difference between those states you can calculate activation energy and the reaction energy the activation energy corresponds to the very height the reaction needs to overcome to go on forward so if the activation energy is too high it's likely that the chemical reaction will not occur if you have multiple pathways and evaluate the activation energy of each reaction then you could investigate which chemical reaction is more favorable than the others you can also create a new chemical reaction pathway on your computer

using quantum chemistry simulation next you can also tackle molecular properties the molecule on your left is the beta-carotene molecule and the figure on the right is the computed absorption spectra of this molecule using quantum chemistry simulations this axis is showing the excitation energy which is energy gap between the ground state and the excited state that which also corresponds to the wavelength of the lights to be absorbed by this molecule and this axis is showing the intensity of the spectrum so you see a large peak here which ranges between purple a violet and green so the

violet blue green lights will be absorbed by this molecule and the remaining yellow orange red lights will be reflected so that's why this beta-carotene's color is orange we can use quantum chemistry simulation to investigate various molecular properties and also we can create a new molecule in our computer and by tuning the properties and functions of the molecule we can create a new molecule with a function of your desire we can also tackle molecular dynamics using quantum chemistry simulation a colleague who was a gas phase experimentalist came to me and said look i have a molecule

here and it forms a hydrogen bond with a water molecule which is hydrogen bond by the way is a weak bond and if this molecule forms a weak bond the hydrogen bond with the water molecule in this location and dislocated so there's two location which forms a hydrogen mod with the water molecule however when i ionize this system the water molecule forms a bond only here so there's only one location so if i start from a neutral state molecule with a water molecule here what will happen if i ionize this system in experiment they found

out that the water molecule starts from here but ends from and ends end up here but they don't know what's happening in between so that's why my colleague came to me and said can you use simulation to investigate what's happening in between so i perform a little dynamic simulation of that system and so first i ionize the system and see how the dynamics will go so the water immediately detaches from here after ionization and it wanders around and forms the water forms a hydrogen bond with a water molecule here and it was in it's a

bit interesting to see that the water always goes in this direction not in this direction and depending on the trajectory it folds around sometimes but it always ends up here so what that means is that using quantum chemistry simulation we could investigate something that could not be observed in experiments so you see that quantum chemistry simulation can be extremely powerful so quantum chemistry acceleration uh is also called electronic structure calculation solves the distribution of electrons and molecules various static properties can be computed using quantum chemistry simulation and also we can understand motions and molecules using

quantum chemistry summations as well the typical static properties we could compute are shown on your right hand side so we can compute equilibrium geometries and we can calculate the barrier heights and investigate reaction pathways of chemical reactions as we did in the first example and also we can compute spectra the second example is showing you the result of the absorption spectra but lita could also compute the nmr chemical shift the circular decreasing spectra the infrared spectra as well we could also compute electronic properties such as dipole moment the polarizabilities and so on and also we

can compute thermodynamic properties so quantum chemists use these quantum quantum chemistry simulation to investigate the properties of molecules or investigate various chemistry problems but not just that also chem researchers doing experiments also use quantum chemistry simulation what they do is that they use quantum chemistry simulation to interpret their experiments especially if you have a structural spectra a data spectra it'll be really convenient if you use quantum chemistry simulation to interpret your spectra so quantum chemistry stimulation is not just important for quantum chemists nowadays it's also important for other chemists as well so you see quantum

chemistry simulation is a very important tool now i'd like to move on to the next section so i'd like to talk about the schrodinger equation and born oppenheimer approximation and i want to show you what kind of equation we want to solve in quantum chemistry simulation first i'd like to show you the time dependent schrodinger equation so this is the time trojan your equation here's the hamiltonian which has the kinetic term and the potential term and psi is the wave function so you can see that the hamiltonian the potential energy term in the hamiltonian and

the wave function they all depend on space r and time t and if you see the left hand side of this equation you see the first derivative of the wave function against time so what this equation is describing is it's describing the quantum system evolving in time so by solving this time dependent structure equation you can solve the propagation of the wave function this time dependent schrodinger equation is a partial differential equation note that we here do not consider relativist effect or spans next i'd like to show you the time independent schrodinger equation so here

is the equation now the hamiltonian which again has a kinetic term and the potential term and the wave function psi only depends on space r it doesn't depend on time anymore we solve this time-dependent schrodinger equation when we're interested in the time independent problem such as the quantities in equilibrium so in time-dependent charging your equation you solve it when you're interested in the time dependent problem of course so in the case of chronic chemistry we're interested in the time independent problem the hamiltonian is time independent so we're going to solve this time-independent schrodinger equation the

eigen this short time index trillion equation is an eigenvalue equation so if we solve this equation we get the eigenvalues as total energy and eigenvectors as the wave functions there's a small n here but what it means it means that by solving the time it depends on the schrodinger equations we could get multiple roots so what that means is that we can calculate the ground state and not just ground sleep but we could also compute the excited states which are above the ground state so we could get multiple routes today however i my talk is

focusing on the ground state so in most of the equations these n are dropped off but please remember that if we can solve the time independent trojan equation we could get multiple roots we could get not just the ground state energy but we could also get the excited state energy as well now i'd like to move on to show you the molecular hamiltonian here we assume that our molecule has two nuclei and and two electrons the nuclei are shown in this large red circle nucleus a with mass m a and char zip a and nucleus

b mass and b it has mass mb and charges it b and the small particles here are the electrons they're labeled with the lowercase letters oh the nucleus are labeled with the upper case letters and electrons i and j here has the mass of small m and charge of minus one so there are five terms in the molecular hamiltonian the first two are the kinetic term and the latter three are the potential term i will look into the terms one by one so the first term here is the kinetic energy of nuclei the upper case

a runs through all the nucleus nuclei and in mar in this molecule it runs from a to b the second term is the kinetic energy of the electrons so the lowercase letter i runs from all all none through all the electrons and in this case it runs to from i to j the third term is the nucleus nucleus repulsion which is shown here the ab runs through all the nucleus nuclei and i'm just this is for to avoid the difficult counting and if you see this this actually the electrostatic potential between the nuclei so four

pi epsilon naught and r a b will be the distance between the two and and on the numerator we have the set a and z b so here you can see that it's the electrostatic potential and the fourth term is electron nucleus attraction what i mean by attraction is that the charge are the opposite so they attract with each other so large a runs through all nuclei and small i runs through all the electrons and you see there's four pi epsilon not r i a with one lower case and one uppercase letters and those interactions

is shown here are i a rj rja rib and rjb they all belong to the electron nucleus attraction again they are the electrostatic potential between electrons and nuclei the last term is the electron electron repulsion in this figure it's shown here again it's 4 pi epsilon not with the distance of rij and its e square so it's the charge of two electrons and this runs through i j runs through all electrons and again avoiding the double counting so these are the five terms of the nuclear molecular hamiltonian now i'd like to introduce the atomic units

in atomic units the reduced plane constant the mass of the electron the charge of the electron and four pi epsilon naught for the electrostatic potential are all one so this hamiltonian i showed you the last slide is now like this it's more simple by the way i use the same ma here and here however it has a different meaning this m a is a relative mass from the electrons in quantum chemistry simulation there are cases that we deal with hamiltonians which has relativistic effect or hamiltonian considering spin orbital coupling however in the talk today i'm

not going to consider relativist effect and spin orbital coupling now i'd like to move on to the born opponent approximation so nuclear mass which is ma here is very large compared to the mass of the electrons so electron motion electronic motion is extremely fast compared to the nuclear motion so now we could have an approximation here so we can assume that the motion of electrons can be decoupled with the motion of nuclei so we could freeze the nuclei and born oppenheimer approximation mostly works in most of the chemical problems however there are exceptions one example

is a conical intersection at conical intersection the ground state and the first excited states are very extremely close in in this insanity the nuclear motion and electron motion couple strongly therefore in this case the born oppenheimer approximation breaks down however in most cases a born oppenheimer approximation works so therefore in frontal chemistry simulation we apply the born oppenheimer approximation so here was the molecular hamiltonian and let's see what happens to the hamiltonian after applying the born oppenheimer approximation so as i mentioned we could freeze the nuclei so the kinetic energy of the nuclei can be

removed and also the nuclei can be treated as classical point charges and the nuclear coordinates will be fixed therefore here the nucleus nucleus impulsion term this will be a completely classical electrostatic potential and this will be a constant so we don't have to explicitly treat this term when we solve the problem it'll be just a simple shift in the energy so then we could determine the electron hamiltonian after the born and hoffenheim approximation so this will be the electronic hamiltonian the first term will be the kinetic energy term of the electrons the second term will

be the electron nucleus attraction and the last term will be the electron electron repulsion so we only have three terms and by the way i've introduced this showing that the nuclear coordinate r will be fixed now i'd like to discuss about the electronic wave function the wave function psi is like this so i mentioned that i was not considering the expense previously but now i'm going to introduce the spins in addition of the spatial coordinates x y z i will introduce a new coordinate here which is omega which is the spin coordinates so for each

electron i have four coordinates okay so x one y one z 1 omega 1 for electron 1 and x 2 y 2 z 2 omega 2 4 electron tail the square of psi is the conditional probability for finding electrons at the given coordinates and this electronic wave function needs to satisfy another condition that is the anti-symmetry against the exchange of electrons so what that means is that if we exchange electron one and extra on two like this here now here we have electron two here and an electron one here it has the chain signs so

this is the anti-symmetry against the exchange of electrons and our wave function have to fulfill this condition now we have the electronic hamiltonian electron wave function so we can determine the electron scoring equation we want to solve so in quantum chemistry simulation we are solving this electronic schrodinger equation so the hamiltonian will be the kinetic term of the electrons the nucleus electron attraction and electron electron repulsion so by solving this schrodinger equation we can compute the electronic energy shown here and here so we have another term actually the nucleus nucleus repulsion term which i mentioned

that it's fully classical so we can calculate separately from this solving this electronic charging equation and add it afterwards as a shift to the energy and the total energy is the sum of this electronic energy and the nucleus nucleus repulsion term of the energy this total energy is also called the potential energy which depends on the nuclear coordinates if you have any experience of running quantum chemistry simulation before maybe you have seen an output showing the electronic energy and the total energy the electronic energy is energy you get when you solve this electron schrodinger equation

and the total energy is the energy adding these two terms so next i'm going to explain what i mean by potential energy depending on the nuclear coordinates so what i mean by the potential energy depend on the nuclear coordinates for instance if i have a system like this a diatomic system and in this molecule we only have one internal degree of freedom which is the inter-atomic distance so suppose i calculate the electron extraordinary equation with one long distance and then i change the bond distance little by little and calculate the electronic throttling occasion again and

again again and if i plot the energy it'll be a smooth curve like this this surface constructed by varying the nuclear coordinate is called the potential energy surface so if you see this potential energy surface the x-axis axis is inter-atomic distance and the y-axis will be the potential energy so you see the potential energy is a function of the interatomic distance here which is actually the nuclear force so that's what i mean by the potential energy depends on the nuclear coordinates and this is the potential energy surface but since we have only one internal degree

of freedom in this molecule which is the interatomic distance we call this a potential energy curve the potential energy curve or surfaces have a lot of important information for instance the minimum of this potential energy curve is showing the most stable point which has the equilibrium geometry here so this is the optimal bond length of this molecule and if you shorten the bond length the atoms will feel the repulsive force and it'll be more the uh the molecule will be more unstable and if we elongate the bond distance then the interaction will be reduced and

at a certain point these two atoms will not feel each other so the energy will converge to an energy of a plus b so the energy difference at the energy of the dissociation and energy at the equilibrium geometry the difference is the bond energy so what that is is that it's the energy the molecule gains when it forms a bond so you see you're going to find a lot of import important information in the potential energy surface so i showed you this slide in the first example of the introduction i vaguely mentioned that this was

the energy but this was actually the potential energy and i just vaguely mentioned that the reaction goes into this direction but it was actually along the reaction coordinate which depends on the nuclear coordinates so i was just showing you the potential energy diagram along the reaction coordinate so if i write a potential energy surface of a reaction i'll get a potential energy surface like this so here is the reactant the most stable point and the transition state the maximum and the product state minimum for all three points they will be at the stationary points and

the potential energy surface what do i mean by the stationary points is that v is the potential energy and its first derivative against each eternal degree of freedom is all zero so they're all zero for these three points and the reactant and the product you can see that it's they're at the minimum of the potential energy surface while the transition state is at the saddle point of potential energy surface the saddle point is that all but one for all degrees of freedom except one it'll be at the minimum while at one degree of freedom like

this it'll be the maximum this is a saddle point and we can distinguish the minimum and the maximum using the hessian matrix the hessian matrix is the second derivative of two internal degrees of freedom and if we construct this hesi matrix and diagonalize it for a minimum we'll get all eigenvalues as positive and at the saddle point we'll get one eigenvalue is negative and the remaining all other eigenvalues as positive so these are the stationary points in the potential energy surface last of all in this section i want to talk about the molecular dynamics simulation

i showed you and the third example of the introduction maybe some of you might thought that the molecular dynamics is dynamic so it's a time dependent problem why aren't you solving the time-dependent schrodinger equation if i was treating the nuclear motion quantum mechanically yes you're right however i am using applying the born oppenheimer approximation since nuclear nucleus i'm only going to treat the classical effect of the nucleus as a point charge so the nuclear motion is classical so therefore i'm not going to solve the time dependent surgery equation i'm just going to simply solve the

classical equation motion however the nuclei are feeling the force from the electrons around them so that's the reason why i want to treat the quantum effect from the electrons therefore i'm i'm performing the quantum chemistry simulation to calculate the force from the electrons on the nuclei and the force of course will depend on the new coordinate of the nuclei the electrons and nuclear motions will be completely decoupled so i was moving the nuclei little by little using molecular dynamics simulation but what i was doing was i was monitoring the dynamics along the potential energy surface

i hope it makes sense to you so i've talked about the schrodinger equation and born oppenheimer approximation in this section i talked about time-dependent strategic equation time independent trojan equation and born oppenheimer approximation now i think you know what kind of equation we want to solve in quantum chemistry simulation we want to solve this electronic shortage equation and i also talked about the potential energy and potential energy surface next i'd like to move on to talk about the quantum chemistry methods i'm going to talk about the hercules fox method semi-empirical methods correlation methods which include

perturbation theory configuration interaction and couple cluster methods and last of all i'm going to talk about density function theory so let me start from the heart defog method so as i mentioned we're going to solve the electronic schrodinger equation and here is electronic hamiltonian which has the kinetic energy of electrons nucleus electron attraction and electron electron repulsion the last term the left electron electron repulsion term is the most difficult term to deal with so we want to make this a bit more simple so it'll be more easier to solve so the underlying idea is the

mean field approximation or we could also call it the independent particle model so of course each electron feel each other in a different mate however we could introduce an approximation such that each electron feel the other electron as an average field if we can introduce that this will become like this so this will be sort of an average potential and it will be one electron operator so as you can see the kinetic term of the electrons and the electron nucleus attraction term they have they're both one electron terms so if we can make this a

one electron term all the terms will be one electrons when it's not terms so it'll be far more easier to solve so hopefully we could do this so this is the underlying idea of the heartrefock method next we'd like to consider the wavefunctions that we're going to use in the archifac method so first we consider describing the wavefunctions as product orbitals so first we have the wavefunction here which has four coordinates x y z first base and omega for spin and we're going to combine all these r omega to x a new coordinate so r

spin orbital has chi and x is coordinate the two particle wave function then can be described as product of orbitals which is psi x1 x2 equals chi 1 x1 times a chi 2 x2 and x we exchange the two electrons it'll be psi x2 x1 equals chi 1 x2 chi 2 x1 but if we have if we describe our wave function as product orbitals these two will be the same so it won't be anti-symmetric against exchange as i mentioned to you our wave function has a satisfied anti-symmetry against exchange so this idea won't work so

we can't treat our wave function as product of orbitals so we need a better strategy instead we could describe our wave function as slater determinants so this is a two particle case and this is a slater determined for the two particles we can calculate either this easily so chi 1 x 1 times chi 2 x 2 minus chi 1 x 2 times chi 2 x 1 like this so it'll be like this and if we exchange the particles now then if you check it you'll see that the sign will be flipped so if we describe

our wave functions as later determinants the anti-symmetry of against exchange will be satisfied so we can write it in a more general form for n particles this will be the slater determinant and from now on i'm going to write this later determinant in this form in hartree we will describe the wave function as a single slater determinant so we want to solve this electronic schroding equation but it is it cannot solve this analytically we have to solve this numerically and we could do so by employing the variational principle so here is the function of the

risk function we want to minimize using the variational principle this is shown here so the numerator here is the expectation value of the hamiltonian the electronic caliphate it could be also written like this and in the denominator what this is it's overlapped with wave functions so if we normalize it it should be one so minimizing this function is equivalent to minimizing this expectation value under the constraint of this being one the variational theorem assumes that the true ground state will be the lower bound so we could what we can do is we can minimize the

energy by burying the parameters or bearing the wave function or optimizing slater determinant so we can minimize this energy by doing so so we can put the trial wave function in it and repeat it until the convergence criteria this will be fulfilled and here is the hartree fog energy expression the first term i include i introduced this new operator which is shown here it is the kinetic energy and electron nuclear attraction attraction for electron i so this integral is actually expectation value of the kinetic energy and the electron nuclear repulsion for electron i so this

term runs for through all the electrons how about these terms this these are the two electron integrals the first term is the coulomb term and the second term exchange integrals so i want to show you what this is these are in the next slide so now i'd like to show you what the two electron integrals are so this the first one is the coulomb integrals so if you see this carefully it's chi i star x1 times chi i x1 so this you could see that this is the density electron eye and again if you see

here it's kyj star x2 and xj akaij x2 which is the density of electron j and we have this one over r operator so what that means is we're calculating the electrostatic potential between electrons i and j i think this is easy to understand because this has a classical counterpart so we have a electrostatic potential with two point charges it's the same as that but here we have a distribution so we're going to calculate digital subtract potential between the densities how about the exchange intervals it's very it's very similar but you see that j here

and i are swapped this exchange integrals uh d arise from the anti-symmetry against exchange so these don't have any classical counterparts they're purely quantum since they originate from exchange oh with the only we only treat exchange drugs with of the same spins here while the coulomb materials we consider both opposite spins and same spins so these were the two electron integrals minimizing the heartrefocus energy which i just showed you with respect to the orbitals we arrive at this harky fog equation so this epsilon is energy or eigenvalue of a spin orbital and chi i is

a forced spin orbital and h is the one electron term which is the kinetic energy of the electrons and electron nucleus attraction and here you see a term here which is showing the electron density of electron j here so you see this is a contribution from the coulomb materials and this is also quite it looks similar but here you see that the spin of i uh the index of i j are swapped here so this should be the contribution from the exchange integrals so i'm going to introduce two new operators j and k which are

here and here this these these three terms will have a more simpler form like this note that how this k operator works is that if the k operator acts on chi i it'll be swapped to become chi j so this is a result of chi j operator acting on chi i that's what it means so and then we have this simple form of equation which we saw in the mean field approximation slide so by using the slate of determinant as a wave function and using the variational principle we arrive at equation just the same we

saw in the mean field approximation so we're indirectly introducing the midfield approximation here so then what that means that it should be easy to solve however it's still difficult at this moment because we have the integrals here and we also have a differential in the kinetic energy term so it is an integral differential equation and it is yet difficult to solve so we have to do something extra so we will introduce another concept which is the linear combinations of atomic orbitals so here is the spin orbitals and we're going to discard these spin orbitals with

the linear combination of atomic orbitals chi bar they're also called the basis sets so what we're going to do here is that we're going to perform or carry out the integrals using the atomic orbitals the it so and if we perform carry out the integrals first then we can get the simple matrix form here so what that means is that by introducing the atomic orbitals we can convert the problem into pseudo eigenvalue problem here the eigenvector are the molecular orbital c they are also known as electro-orbital coefficients and we also get eigenvalues epsilon here which

are the molecular orbital energies so now we could have we now we finally have the equation that we can solve easily and by the way we can't solve this in one shot because if you remember in f we have this chi in the j and k operators so what that means is we have c also in f so we need to solve this iteratively so that's our church of procedure is called self-consistent field procedure we start from the input of the 3d coordinates the atomic nuclei as we mentioned the potential energy depends on the coordinates

of the nuclei and we also determine what kind of atomic orbitals we'll use then we will construct the initial guess of the molecular orbitals and afterwards we construct the fog matrix and then we diagonalize it so at this point we get a new c the more likely or orbital coefficients and if if it's converge we terminate the calculation and if not we go back to three and repeat this process until the convergence is fulfilled so at the end we get optimized molecular orbitals and hartree-fock energy so what we mean by molecular orbitals are optimized is

that we get the lyric coefficient of the linear combination to form a molecular orbitals using the linear combination of atomic orbitals so we can construct this the molecular orbitals here and here using these atomic orbitals and c is showing the suitable coefficients for the linear combination so here we have uh talked about the hard defog method so in hartree we use a single slater determined as the wave function the variational principle will be used to calculate the minimized energy the parameters will be in the wave function and we'll we'll optimize the parameters to get suitable

slater determinant this introduces the mean field approximation where each electrons field other electrons as an average field we also introduced another concept the linear combination of atomic orbitals which helps our problem become a pseudo eigenvalue problem and i also explained about self-consistent field procedure we use to perform heartry calculation so now i'd like to move on to the uh next part of my talk so i'm going to talk about the basic sets we use in quantum chemistry simulation and it's atomic orbital integrals that we compute in quantum chemistry simulation so first i'd like to talk

about the basis sets what are the basis sets the basis sets are the functions for representing the atomic orbitals that describe the electronic wave function so we construct the molecular orbitals using these atomic orbitals there's several types of biases sets the first one is the gaussian basis sets they're frequently used in quantum chemistry calculations for isolated molecules the second one is the plane wave basis set they're used in calculations for extended system such as crystals metal semi semiconductors there's also a real base space basis such as finite element and wavelets in quantum chemistry simulation we

mostly use the gaussian bases so today i'm going to talk about the gaussian base assets and if we can increase the number of gaussian base assets we get more accuracy in our calculation however we are increasing the computational cost so it's the accuracy versus computational cost trade-off so here is the general form of the gaussian base gaussian function shown here and here i'm not using chi because they're simply spatial orbitals they don't depend on spins so i have phi m psi so we're gonna get we're gonna form the spatial orbitals psi using the linear combination

of these fives so this is the gaussian function and x one i y i and z i are the center of the gaussian function typically we choose the center of the nuclei as the center of the gaussian function and in gaussian basis sets the predefined parameters are this and this so they're determined based on the atomic properties so it only has information of that it doesn't have any other empirical information in it and as i mentioned if we have more number of basis sets we can get our computational results more accurate however we don't want

to increase the basis sets as much as possible because we're increasing the computational cost so we want to make our bases more flexible but we do not want to increase the number of c what can we do we can introduce this contracted basis so we only have one's phi this one the number five won't change but we can define this as a linear combination of a couple of gaussian functions so in this case also dij is a predetermined parameter based on atoms by doing so we could increase the number of bases while not increasing the

number of the coefficients to optimize there are several different type of phases sets one of them are the minimal basis sets it's using the minimal number required to represent all electrons on atoms the famous one is sto 3g basis set the next one is a split valence base assets it represents the valence orbitals by more than one function while it treats the core orbitals using one function so in chemistry problems the valence orbitals play an important role compared to the core orbitals so that's why we won't use more functions for the valence orbitals while you

won't use much for the core orbitals so that's the idea of sphalen's basis sets the famous ones are the purple bases sets and the 631g star or like six three one one plus gg g star star and what does star mean star is representing the polarization functions which have the higher angular momentum i'll talk about this later but what's the difference between star and star star means that the first star is means that it's going to apply the polarization functions for the heavy atoms such as the atoms in the second row like carbon or nitrogen

or oxygen and if you have star star the second one means that you're gonna you're gonna introduce the polarization functions to light atoms which is hydrogen we could also we also have a term plus which is means the diffuse function the diffuse functions are functions which have the small alpha here so we have more of a widely spread distribution these are frequently used in molecules such as anions where they have more of a widely spread distribution compared to neutral species there are also correlation consistent basis sets which are systematic basis sets for conversion correlation methods

to get very accurate results the famous ones are the ccp v and z so what ccp v and z means is it's if it's d c c p v d z it means that you're using two functions per ao atomic orbital if it's t z you're using three functions for each atomic orbital and if it's q z it'll be four and if p5z it'll be using five functions for each atomic orbital yeah these are the uh quite famous ones in quantum chemistry so i also want to explain what sdo ng or 631 means so here

is the form here is the form of the contracted basis sorry i mis typed here this should be around here but uh it's okay so first about the minimal basis sets there's an expression called s-t-o-n-g what that means is that we are trying to make a one slater type orbital so we're going to make one orbital here with using n gaussian functions so we want to use n of these functions for making this one this function so sdo3c again means we're going to build this one orbital of one base set using n number of contracted

basis sets and for the split valence spaces there is the term like 6 3 1 or 6 3 one one what is that the first six is describing the number of contracted bases for our core atomic orbitals so in split venus basis sets we only use one function one of this to describe the core atomic orbitals but we're going to use six of these describe one of this that's what six means and how about the next numbers 3 1 or 3 1 1 what that means so 3 1 has two numbers and 3 1 1

has three numbers so what that means is that for the former we're going to use two bases to describe the valence atomic orbitals and if it's three and one we're going to use three basis sets three to describe the valence atomic orbitals so that's what the number what's that's what's up three one or three three one means but how about what actually three or one means it means that we're going to use two bases as as we mentioned but we're going to use three of these for the first one and one of these for the

second one and for three one one that means we're going to use three of these contracted functions for the first one one function for the second one and one function for the third one that's what 3 1 or 3 1 1 means i hope you understand what these mean now now i'd like to move on to angular momentum i mentioned that there are cases that we need to use basis sets for higher angular momentum so i want to discuss what i meant by that so here is the basis that we have and l x l

y l z or the angular momentum for each x y and z so here this figure is the solution of the strontium equation of hydrogen atom these are the hydrogen orbitals or the atomic orbitals you get by solving the schrodinger equation of a hydrogen atom and for l equals zero you have this s-type function for l equals one you have a p-type solution and for l equal two you have a d type function in l equal three you have f-type functions and as the number of l increases the number of nodal planes increases as well

so here you have one here you have two node planes and here you have three nodal planes so if we are dealing with hydrogen for instance we usually only need s orbitals because we only have s orbitals filled in hydrogen however what we want to do here is that we want to describe the molecular orbitals accurately as possible using the atomic orbitals so in that sense sometimes using atomic orbitals of higher quantum numbers will help describing the molecular orbitals accurately so that is the reason why we use p functions or d functions even d functions

for hydrogen as well so we use orbitals of higher angular momentum to make our results accurate as possible now i'd like to move on to show you why we're using gaussian functions so here's one of the advantage using a gaussian function so first advantage is the gaussian product theorem suppose we have two gaussians in one d so we can combine this to make it to a one gaussian function with a pre-factor if you see the pre-factor and if you look at mu and x you see they are all fixed values so this will be a

constant so and so we could we have made one two gaussians into one so that's one advantage and the second advantage is the recursion relation so here's the general form of the gaussian function but x y and z are separable so why don't we just focus on x so x is the form like this of course by definition this is quite obvious so gl times x minus x should be of course gl plus 1. but what this means is that if we calculate g with lower angular momentum we could use the recursion relation to compute

the g of the higher angular momentum and also if we once calculated g then we can easily calculate the gradients shown in this recursion relation form so what this means that we can calculate both gradients and also the gaussians with higher angular momentum easily using the recursion relation so this recursive relation is the second advantage so now i'd like to move on to talk about integrals computed in quantum chemistry we're going to talk about integrals in atomic orbitals so again this will be the gaussian atomic orbital gaussian basis set and we're going to use this

to compute the one electron integrals and two electron integrals so the one and electron integrals has the electron a kinetic energy term of the electrons and electron nucleus attraction term and it's two two electron integrals have the electron electron repulsion term and as an example we'll try to compute this so let us assume that we we could start from the very simple gaussian function so we should have a value of l x l y and l z but let's start by considering the simplest gaussian function which where l x and l y l c equals

zero so this will be a very simple gaussian and we are going to apply the same thing to all gaussian functions mu lambda and sigma and then we can use the gaussian product theorem and combine all these gaussian functions and then we'll get this equation so the first part here 2 pi is something k1 and k2 k1 if you see this k1 will be depending only on the parameters fixed constants of mu and u is shown here so this will be all constant so k1 will be a constant value and again k2 depending on the

parameters the constant values of lambda and sigma so again this will be the constant value and same goes for p and q p and q so now they're all constants so this is a constant term in the help of this term so f0 alpha rpq squared the general form of this function is the incomplete gamma function shown here so this is the function we will want to evaluate the integrals and by evaluating this integral we can compute this integral and once we get this then we could get the integral value of this function using the

recursion relations so we're going to first use the gaussian product theorem and make it simple to compute and then we're going to calculate incomplete gamma function and then we're going to use the recursion relations to get the values of high angular momentum so i talked about the basic sets and atomic orbitals orbital integrals so first i mentioned about the base sets and i focus on the gaussian basis sets and i talked about the advantage using gaussian basis sets the gaussian product theorem and the recursion relations i also talked about atomic orbital integrals and i just

very briefly show an example of how we calculate the two electron integrals so this is the last slide of the first half of my talk there is a term called ab initial molecular orbital methods so i showed you the gaussian basis sets and i mentioned about the predefined parameters based on atoms besides that we have no empirical parameters in our calculation so therefore we call it the initial molecular orbital methods from the start and they include the harpy fog method which i showed you and perturbation theory and configuration interaction and couple cluster methods which i'm

going to show you later on these are the initial molecular orbital methods and on the other hand semi-empirical methods include empirical parameters so they are not have initial methods and there's also density functional theory and it's first principle but and some people actually even say that it's evidential but it's administ it's not advantageous or orbital methods it's a completely different concept so well i think it's fair not to include density function theory in ab initial molecular orbital methods okay so i guess this is the midpoint of my talk so let us take a break and

i'll stop sharing my screen and i will pause the recording let us have a break