Slides from FYS4411 Lectures

Transcription

1 1 / 95 Slides from FYS4411 Lectures Morten Hjorth-Jensen & Gustav R. Jansen 1 Department of Physics and Center of Mathematics for Applications University of Oslo, N-0316 Oslo, Norway Spring 2012

2 2 / 95 Topics for Weeks 10-15, March 5 - April 15 Slater determinant, minimization and programming strategies Many electrons and Slater determinant How to implement the Slater determinant Minimizing the energy expectation value (Conjugate gradient method) Optimization Project work: Project 1 should be about done before the end of the period. Project 2 will be presented after easter.

3 3 / 95 Slater determinants Φ(r 1, r 2,..., r N, α, β,..., σ) = 1 N! ψ α(r 1 ) ψ α(r 2 ) ψ α(r N ) ψ β (r 1 ) ψ β (r 2 ) ψ β (r N ) ψ σ(r 1 ) ψ σ(r 2 ) ψ γ(r N ) where r i stand for the coordinates and spin values of a particle i and α, β,..., γ are quantum numbers needed to describe remaining quantum numbers., (1)

4 4 / 95 Slater determinants The potentially most time-consuming part is the evaluation of the gradient and the Laplacian of an N-particle Slater determinant. We have to differentiate the determinant with respect to all spatial coordinates of all particles. A brute force differentiation would involve N d evaluations of the entire determinant which would even worsen the already undesirable time scaling, making it Nd O(N 3 ) O(d N 4 ). This poses serious hindrances to the overall efficiency of our code. The efficiency can be improved however if we move only one electron at the time. The Slater determinant matrix D is defined by the matrix elements d ij φ j (x i ) (2) where φ j (r i ) is a single particle wave function. The columns correspond to the position of a given particle while the rows stand for the various quantum numbers.

5 5 / 95 Slater determinants What we need to realize is that when differentiating a Slater determinant with respect to some given coordinate, only one row of the corresponding Slater matrix is changed. Therefore, by recalculating the whole determinant we risk producing redundant information. The solution turns out to be an algorithm that requires to keep track of the inverse of the Slater matrix. Let the current position in phase space be represented by the (N d)-element vector r old and the new suggested position by the vector r new. The inverse of D can be expressed in terms of its cofactors C ij and its determinant D : d 1 ij = C ji D (3) Notice that the interchanged indices indicate that the matrix of cofactors is to be transposed.

6 6 / 95 Slater determinants If D is invertible, then we must obviously have D 1 D = 1, or explicitly in terms of the individual elements of D and D 1 : NX k=1 d ik d 1 kj = δ ij (4) Consider the ratio, which we shall call R, between D(r new ) and D(r old ). By definition, each of these determinants can individually be expressed in terms of the ith row of its cofactor matrix P N R D(rnew ) D(r old ) = j=1 d ij(r new ) C ij (r new ) P N j=1 d (5) ij(r old ) C ij (r old )

7 7 / 95 Slater determinants Suppose now that we move only one particle at a time, meaning that r new differs from r old by the position of only one, say the ith, particle. This means that D(r new ) and D(r old ) differ only by the entries of the ith row. Recall also that the ith row of a cofactor matrix C is independent of the entries of the ith row of its corresponding matrix D. In this particular case we therefore get that the ith row of C(r new ) and C(r old ) must be equal. Explicitly, we have: C ij (r new ) = C ij (r old ) j {1,..., N} (6)

8 8 / 95 Slater determinants Inserting this into the numerator of eq. (5) and using eq. (3) to substitute the cofactors with the elements of the inverse matrix, we get: P N j=1 R = d ij(r new ) C ij (r old P N ) P N j=1 d ij(r old ) C ij (r old ) = j=1 d ij(r new ) d 1 (r ji old ) P N j=1 d ij(r old ) d 1 (r ji old ) (7)

9 9 / 95 Slater determinants Now by eq. (4) the denominator of the rightmost expression must be unity, so that we finally arrive at: R = NX j=1 d ij (r new ) d 1 ji (r old ) = NX j=1 φ j (r new i ) d 1 (r old ) (8) ji What this means is that in order to get the ratio when only the ith particle has been moved, we only need to calculate the dot product of the vector `φ1 (r new ),..., φ i N (r new ) of single particle wave functions evaluated at this new i position with the ith column of the inverse matrix D 1 evaluated at the original position. Such an operation has a time scaling of O(N). The only extra thing we need to do is to maintain the inverse matrix D 1 (x old ).

10 10 / 95 Slater determinants The scheme is also applicable for the calculation of the ratios involving derivatives. It turns out that differentiating the Slater determinant with respect to the coordinates of a single particle r i changes only the ith row of the corresponding Slater matrix.

11 11 / 95 Slater determinants The gradient and Laplacian can therefore be calculated as follows: i D(r) D(r) = NX j=1 i d ij (r) d 1 ji (r) = NX j=1 i φ j (r i ) d 1 ji (r) (9) and 2 i D(r) NX = 2 i D(r) d ij(r) d 1 (r) = ji j=1 NX j=1 2 i φ j(r i ) d 1 ji (r) (10)

12 12 / 95 Slater determinants If the new position r new is accepted, then the inverse matrix can by suitably updated by an algorithm having a time scaling of O(N 2 ). This algorithm goes as follows. First we update all but the ith column of D 1. For each column j i, we first calculate the quantity: S j = (D(r new ) D 1 (r old )) ij = The new elements of the jth column of D 1 are then given by: NX l=1 d il (r new ) d 1 lj (r old ) (11) d 1 kj (r new ) = d 1 (r old ) S j kj R d 1 (r old ) ki k {1,..., N} j i (12)

13 13 / 95 Slater determinants Finally the elements of the ith column of D 1 are updated simply as follows: d 1 (r new ) = 1 ki R d 1 (r old ) k {1,..., N} (13) ki We see from these formulas that the time scaling of an update of D 1 after changing one row of D is O(N 2 ).

14 14 / 95 Slater determinants Thus, to calculate all the derivatives of the Slater determinant, we only need the derivatives of the single particle wave functions ( i φ j (r i ) and 2 i φ j(r i )) and the elements of the corresponding inverse Slater matrix (D 1 (r i )). A calculation of a single derivative is by the above result an O(N) operation. Since there are d N derivatives, the time scaling of the total evaluation becomes O(d N 2 ). With an O(N 2 ) updating algorithm for the inverse matrix, the total scaling is no worse, which is far better than the brute force approach yielding O(d N 4 ). Important note: In most cases you end with closed form expressions for the single-particle wave functions. It is then useful to calculate the various derivatives and make separate functions for them.

15 15 / 95 Slater determinant: Explicit expressions for various Atoms, beryllium The Slater determinant takes the form Φ(r 1, r 2,, r 3, r 4, α, β, γ, δ) = 1 ψ 100 (r 1 ) ψ 100 (r 2 ) ψ 100 (r 3 ) ψ 100 (r 4 ) ψ 100 (r 1 ) ψ 100 (r 2 ) ψ 100 (r 3 ) ψ 100 (r 4 ) 4! ψ 200 (r 1 ) ψ 200 (r 2 ) ψ 200 (r 3 ) ψ 200 (r 4 ). ψ 200 (r 1 ) ψ 200 (r 2 ) ψ 200 (r 3 ) ψ 200 (r 4 ) The Slater determinant as written is zero since the spatial wave functions for the spin up and spin down states are equal. But we can rewrite it as the product of two Slater determinants, one for spin up and one for spin down.

16 16 / 95 Slater determinant: Explicit expressions for various Atoms, beryllium We can rewrite it as Φ(r 1, r 2,, r 3, r 4, α, β, γ, δ) = Det (1, 2)Det (3, 4) Det (1, 3)Det (2, 4) Det (1, 4)Det (3, 2) + Det (2, 3)Det (1, 4) Det (2, 4)Det (1, 3) where we have defined and The total determinant is still zero! +Det (3, 4)Det (1, 2), Det (1, 2) = 1 2 ψ 100 (r 1 ) ψ 100 (r 2 ) ψ 200 (r 1 ) ψ 200 (r 2 ) Det (3, 4) = 1 2 ψ 100 (r 3 ) ψ 100 (r 4 ) ψ 200 (r 3 ) ψ 200 (r 4 ),.

17 17 / 95 Slater determinant: Explicit expressions for various Atoms, beryllium We want to avoid to sum over spin variables, in particular when the interaction does not depend on spin. It can be shown, see for example Moskowitz and Kalos, Int. J. Quantum Chem. 20 (1981) 1107, that for the variational energy we can approximate the Slater determinant as Φ(r 1, r 2,, r 3, r 4, α, β, γ, δ) Det (1, 2)Det (3, 4), or more generally as Φ(r 1, r 2,... r N ) Det Det, where we have the Slater determinant as the product of a spin up part involving the number of electrons with spin up only (3 for six-electron QD and 6 in 12-electron QD) and a spin down part involving the electrons with spin down. This ansatz is not antisymmetric under the exchange of electrons with opposite spins but it can be shown that it gives the same expectation value for the energy as the full Slater determinant. As long as the Hamiltonian is spin independent, the above is correct. Exercise for next week: convince yourself that this is correct.

18 18 / 95 Slater determinants We will thus factorize the full determinant D into two smaller ones, where each can be identified with and respectively: D = D D (14) The combined dimensionality of the two smaller determinants equals the dimensionality of the full determinant. Such a factorization is advantageous in that it makes it possible to perform the calculation of the ratio R and the updating of the inverse matrix separately for D and D : D new D old = D new D old D new D old (15)

19 19 / 95 Slater determinants This reduces the calculation time by a constant factor. The maximal time reduction happens in a system of equal numbers of and particles, so that the two factorized determinants are half the size of the original one. Consider the case of moving only one particle at a time which originally had the following time scaling for one transition: O R (N) + O inverse (N 2 ) (16) For the factorized determinants one of the two determinants is obviously unaffected by the change so that it cancels from the ratio R.

20 20 / 95 Slater determinants Therefore, only one determinant of size N/2 is involved in each calculation of R and update of the inverse matrix. The scaling of each transition then becomes: O R (N/2) + O inverse (N 2 /4) (17) and the time scaling when the transitions for all N particles are put together: O R (N 2 /2) + O inverse (N 3 /4) (18) which gives the same reduction as in the case of moving all particles at once.

21 21 / 95 Updating the Slater matrix Computing the ratios discussed above requires that we maintain the inverse of the Slater matrix evaluated at the current position. Each time a trial position is accepted, the row number i of the Slater matrix changes and updating its inverse has to be carried out. Getting the inverse of an N N matrix by Gaussian elimination has a complexity of order of O(N 3 ) operations, a luxury that we cannot afford for each time a particle move is accepted. We will use the expression 8 >< d 1 (x d 1 (x new kj old ) d 1 (x old ) ki R ) = kj >: d 1 (x old ) ki R P N l=1 d il(x new )d 1 lj (x old ) if j i P N l=1 d il(x old )d 1 lj (x old ) if j = i (19)

22 22 / 95 Updating the Slater matrix This equation scales as O(N 2 ). The evaluation of the determinant of an N N matrix by standard Gaussian elimination requires O(N 3 ) calculations. As there are Nd independent coordinates we need to evaluate Nd Slater determinants for the gradient (quantum force) and Nd for the Laplacian (kinetic energy). With the updating algorithm we need only to invert the Slater determinant matrix once. This can be done by standard LU decomposition methods.

23 23 / 95 Slater Determinant and VMC Determining a determinant of an N N matrix by standard Gaussian elimination is of the order of O(N 3 ) calculations. As there are N d independent coordinates we need to evaluate Nd Slater determinants for the gradient (quantum force) and N d for the Laplacian (kinetic energy) With the updating algorithm we need only to invert the Slater determinant matrix once. This is done by calling standard LU decomposition methods.

24 24 / 95 How to compute the Slater Determinant If you choose to implement the above recipe for the computation of the Slater determinant, you need to LU decompose the Slater matrix. This is described in chapter 4 of the lecture notes. You need to call the function ludcmp in lib.cpp. You need to transfer the Slater matrix and its dimension. You get back an LU decomposed matrix.

25 25 / 95 LU Decomposition The LU decomposition method means that we can rewrite this matrix as the product of two matrices B and C where 0 a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 1 C A = b b 31 b b 41 b 42 b C B c 11 c 12 c 13 c 14 0 c 22 c 23 c c 33 c c 44 The matrix A R n n has an LU factorization if the determinant is different from zero. If the LU factorization exists and A is non-singular, then the LU factorization is unique and the determinant is given by det{a} = c 11 c c nn. 1 C A.

26 26 / 95 How should we structure our code? What do you think is reasonable to split into subtasks defined by classes? Single-particle wave functions? External potentials? Operations on r ij and the correlation function? Mathematical operations like the first and second derivative of the trial wave function? How can you split the derivatives into various subtasks? Matrix and vector operations? Your task is to figure out how to structure your code in order to compute the Slater determinant for the six electron dot. This should be compared with the brute force case. Do not include the correlation factor in the first attempt nor the electron-electron repulsion.

27 27 / 95 A useful piece of code, distances double r i ( double, i n t ) ; / / distance between nucleus and e l e c t r o n i double r i j ( double, int, i n t ) ; / / distance between e l e c t r o n s i and j You should also make functions for the single-particle wave functions, their first and second derivatives as well.

28 The function to set up a determinant / / Determinant f u n c t i o n double determinant ( double A, i n t dim ) { i f ( dim == 2) return A [ 0 ] [ 0 ] A [ 1 ] [ 1 ] A [ 0 ] [ 1 ] A [ 1 ] [ 0 ] ; double sum = 0; for ( i n t i = 0; i < dim ; i ++) { double sub = new double [ dim 1]; for ( i n t j = 0; j < i ; j ++) sub [ j ] = &A [ j ] [ 1 ] ; for ( i n t j = i +1; j < dim ; j ++) sub [ j 1] = &A [ j ] [ 1 ] ; i f ( i % 2 == 0) sum += A [ i ] [ 0 ] determinant ( sub, dim 1) ; else sum = A [ i ] [ 0 ] determinant ( sub, dim 1) ; } delete [ ] sub ; } return sum ; 28 / 95

29 Set up the Slater determinant N is the number of electrons and N2 is half the number of electrons. / / Slater determinant double s l a t e r ( double R, double alpha, double N, double N2) { double DUp = ( double ) matrix (N2, N2, sizeof ( double ) ) ; double DDown = ( double ) matrix (N2, N2, sizeof ( double ) ) ; for ( i n t i = 0; i < N2 ; i ++) { for ( i n t j = 0; j < N2 ; j ++) { DUp[ i ] [ j ] = phi ( j,r, i, alpha ) ; DDown [ i ] [ j ] = phi ( j,r, i +N2, alpha ) ; } } / / Returns product of spin up and spin down dets double det = determinant (DUp, N2) determinant ( DDown, N2) ; f r e e m a t r i x ( ( void ) DUp) ; f r e e m a t r i x ( ( void ) DDown) ; 29 / 95

30 30 / 95 Jastrow factor / / Jastrow f a c t o r double j a s t r o w ( double R, double beta, double N, double N2) { double arg = 0; for ( i n t i = 1; i < N; i ++) for ( i n t j = 0; j < i ; j ++) i f ( ( i < N2 && j < N2) ( i >= N2 && j >= N2 ) ) { double r i j = r i j (R, i, j ) ; arg += r i j / (1+ beta r i j ) ; / / same spin } else { double r i j = r i j (R, i, j ) ; arg += 1.0 r i j / (1+ beta r i j ) ; / / opposite spin } return exp ( arg ) ; }

31 31 / 95 / / Check of s i n g u l a r i t y at R = 0 bool S i n g u l a r i t y ( double R, i n t N) { for ( i n t i = 0; i < N; i ++) i f ( r i (R, i ) < 1e 10) return true ; } for ( i n t i = 0; i < N 1; i ++) for ( i n t j = i +1; j < N; j ++) i f ( r i j (R, i, j ) < 1e 10) return true ; return false ;

32 32 / 95 Efficient calculations of wave function ratios The expectation value of the kinetic energy expressed in atomic units for electron i is K b i = 1 Ψ 2 i Ψ, (20) 2 Ψ Ψ K i = i Ψ Ψ. (21) 2 Ψ Ψ = 2 (Ψ D Ψ C ) Ψ D Ψ C = [ (Ψ D Ψ C )] Ψ D Ψ C = [Ψ C Ψ D + Ψ D Ψ C ] Ψ D Ψ C = Ψ C Ψ D + Ψ C 2 Ψ D + Ψ D Ψ C + Ψ D 2 Ψ C Ψ D Ψ C (22) 2 Ψ Ψ = 2 Ψ D Ψ D + 2 Ψ C Ψ C + 2 Ψ D Ψ D Ψ C Ψ C (23)

33 33 / 95 Summing up: Bringing it all together, Local energy The second derivative of the Jastrow factor divided by the Jastrow factor (the way it enters the kinetic energy) is» 2 Ψ C NX k 1 X 2 g = 2 ik Ψ C x x 2 k=1 i=1 k + 0 NX k 1 g ik x k=1 i=1 k NX i=k+1 g ki x i 12 A But we have a simple form for the function, namely 8 9 Ψ C = Y < X ar = ij exp f(r ij ) = exp : 1 + βr i<j i<j ij ;, and it is easy to see that for particle k we have 2 k Ψ C = X (r k r i )(r k r j ) f (r ki )f (r kj ) + X Ψ C r ij k ki r kj j k! f (r kj ) + 2 f (r kj ) r kj

34 34 / 95 Bringing it all together, Local energy Using f(r ij ) = ar ij 1 + βr ij, and g (r kj ) = dg(r kj )/dr kj and g (r kj ) = d 2 g(r kj )/dr 2 kj we find that for particle k we have 2 k Ψ C = X (r k r i )(r k r j ) a a Ψ C r ij k ki r kj (1 + βr ki ) 2 (1 + βr kj ) 2 +X j k! 2a r kj (1 + βr kj ) 2 2aβ (1 + βr kj ) 3

35 35 / 95 Local energy The gradient and Laplacian can be calculated as follows: i D(r) D(r) = NX j=1 i d ij (r) d 1 ji (r) = NX j=1 i φ j (r i ) d 1 (r) ji and 2 i D(r) NX = 2 i D(r) d ij(r) d 1 (r) = ji j=1 NX j=1 2 i φ j(r i ) d 1 (r) ji

36 Local energy function double E l o c a l ( double R, double alpha, double beta, i n t N, double F, double DinvUp, double DinvDown, i n t N2, double detgrad, double j a s t g r a d ) { / / K i n e t i c energy double k i n e t i c = 0; / / Determinant p a r t for ( i n t i = 0; i < N; i ++) { for ( i n t j = 0; j < dimension ; j ++) { } i f ( i < N2) for ( i n t l = 0; l < N2 ; l ++) k i n e t i c = p h i d e r i v 2 ( l, R, i, j, alpha ) DinvUp [ l ] [ i ] ; else for ( i n t l = 0; l < N2 ; l ++) k i n e t i c = p h i d e r i v 2 ( l, R, i, j, alpha ) DinvDown [ l ] [ i N2 ] ; 36 / 95

37 37 / 95 Jastrow part / / Jastrow p a r t double r i j, a ; for ( i n t i = 0; i < N; i ++) { for ( i n t j = 0; j < dimension ; j ++) { k i n e t i c = j a s t g r a d [ i ] [ j ] j a s t g r a d [ i ] [ j ] ; } } for ( i n t i = 0; i < N 1; i ++) { for ( i n t j = i +1; j < N; j ++) { i f ( ( j < N2 && i < N2) ( j >= N2 && i >= N2 ) ) a = ; else a = 1. 0 ; r i j = r i j (R, i, j ) ; k i n e t i c = 4 a / ( r i j pow(1+ beta r i j, 3 ) ) ; } }

38 38 / 95 Local energy / / I n t e r f e r e n c e p a r t for ( i n t i = 0; i < N; i ++) { for ( i n t j = 0; j < dimension ; j ++) { k i n e t i c = 2 detgrad [ i ] [ j ] j a s t g r a d [ i ] [ j ] ; } } k i n e t i c =. 5 ;

39 39 / 95 / / P o t e n t i a l energy / / electron nucleus p o t e n t i a l double p o t e n t i a l = 0; for ( i n t i = 0; i < N; i ++) p o t e n t i a l = Z / r i (R, i ) ; } / / electron e l e c t r o n p o t e n t i a l for ( i n t i = 0; i < N 1; i ++) for ( i n t j = i +1; j < N; j ++) p o t e n t i a l += 1 / r i j (R, i, j ) ; return p o t e n t i a l + k i n e t i c ;

40 40 / 95 Determinant part in quantum force The gradient for the determinant is i D(r) D(r) = NX j=1 i d ij (r) d 1 ji (r) = NX j=1 i φ j (r i ) d 1 (r). ji

41 Quantum force void calcqf ( double R, double F, double alpha, double beta, i n t N, double DinvUp, double DinvDown, i n t N2, double detgrad, double j a s t g r a d ) { double sum ; / / Determinant p a r t for ( i n t i = 0; i < N; i ++) { for ( i n t j = 0; j < dimension ; j ++) { sum = 0; i f ( i < N2) for ( i n t l = 0; l < N2 ; l ++) sum += p h i d e r i v ( l,r, i, j, alpha ) DinvUp [ l ] [ i ] ; else for ( i n t l = 0; l < N2 ; l ++) sum += p h i d e r i v ( l,r, i, j, alpha ) DinvDown [ l ] [ i N2 ] ; detgrad [ i ] [ j ] = sum ; } 41 / 95

42 42 / 95 Jastrow gradient in quantum force We have 8 9 Ψ C = Y < X ar = ij g(r ij ) = exp : 1 + βr i<j i<j ij ;, the gradient needed for the quantum force and local energy is easy to compute. We get for particle k k Ψ C = X r kj a Ψ C r j k kj (1 + βr kj ) 2, which is rather easy to code. Remember to sum over all particles when you compute the local energy.

43 43 / 95 Jastrow part / / Jastrow p a r t double r i l, a ; for ( i n t i = 0; i < N; i ++) { for ( i n t j = 0; j < dimension ; j ++) { sum = 0; for ( i n t l = 0; l < N; l ++) { i f ( l! = i ) { i f ( ( l < N2 && i < N2) ( l >= N2 && i >= N2) ) a = ; else a = 1. 0 ;

44 44 / 95 } r i l = r i j (R, i, l ) ; sum += (R[ i ] [ j ] R[ l ] [ j ] ) a / ( r i l pow(1+ beta r i l, 2 ) ) ; } } j a s t g r a d [ i ] [ j ] = sum ; } } for ( i n t i = 0; i < N; i ++) for ( i n t j = 0; j < dimension ; j ++) F [ i ] [ j ] = 2 ( detgrad [ i ] [ j ] + j a s t g r a d [ i ] [ j ] ) ;

45 45 / 95 Metropolis-Hastings part / / I n i t i a l i z e p o s i t i o n s double R = ( double ) matrix (N, dimension, sizeof ( double ) ) ; for ( i n t i = 0; i < N; i ++) for ( i n t j = 0; j < dimension ; j ++) R[ i ] [ j ] = gaussian deviate (&idum ) ; i n t N2 = N/ 2 ; / / dimension of S l a t e r matrix

46 46 / 95 Metropolis Hastings part We need to compute the ratio between wave functions, in particular for the Slater determinants. R = NX j=1 d ij (r new ) d 1 ji (r old ) = NX j=1 φ j (r new i ) d 1 (r old ) ji What this means is that in order to get the ratio when only the ith particle has been moved, we only need to calculate the dot product of the vector `φ1 (r new ),..., φ i N (r new ) of single particle wave functions evaluated at this new i position with the ith column of the inverse matrix D 1 evaluated at the original position. Such an operation has a time scaling of O(N). The only extra thing we need to do is to maintain the inverse matrix D 1 (x old ).

47 47 / 95 Jastrow factor in Metropolis Hastings We have R C = Ψnew C Ψ cur C = eunew e Ucur = e U, (24) where k 1 X U = i=1 `f new ik fik cur + NX i=k+1 `f new ki fki cur One needs to develop a special algorithm that runs only through the elements of the upper triangular matrix g and have k as an index. (25)

48 48 / 95 Metropolis-Hastings part / / I n i t i a l i z e inverse S l a t e r matrices f o r spin up and spin down double DinvUp = ( double ) matrix (N2, N2, sizeof ( double ) ) ; double DinvDown = ( double ) matrix (N2, N2, sizeof ( double ) ) ; for ( i n t i = 0; i < N2 ; i ++) { for ( i n t j = 0; j < N2 ; j ++) { DinvUp [ i ] [ j ] = phi ( j,r, i, alpha ) ; DinvDown [ i ] [ j ] = phi ( j,r, i +N2, alpha ) ; } } inverse ( DinvUp, N2) ; inverse ( DinvDown, N2) ;

49 49 / 95 Metropolis-Hastings part / / Inverse S l a t e r matrix i n new p o s i t i o n double DinvUp new = ( double ) matrix (N2, N2, sizeof ( double ) ) ; double DinvDown new = ( double ) matrix (N2, N2, sizeof ( double ) ) ; for ( i n t i = 0; i < N2 ; i ++) { for ( i n t j = 0; j < N2 ; j ++) { DinvUp new [ i ] [ j ] = DinvUp [ i ] [ j ] ; DinvDown new [ i ] [ j ] = DinvDown [ i ] [ j ] ; } }

50 50 / 95 Metropolis-Hastings part / / Gradients of determinant and and Jastrow f a c t o r double detgrad = ( double ) matrix (N, dimension, sizeof ( double ) ) ; double j a s t g r a d = ( double ) matrix (N, dimension, sizeof ( double ) ) ; double detgrad new = ( double ) matrix (N, dimension, sizeof ( double ) ) ; double jastgrad new = ( double ) matrix (N, dimension, sizeof ( double ) ) ; / / I n i t i a l i z e quantum f o r c e double F = ( double ) matrix (N, dimension, sizeof ( double ) ) ; calcqf (R, F, alpha, beta,n, DinvUp, DinvDown, N2, detgrad, j a s t g r a d ) ;

51 51 / 95 Metropolis-Hastings part double EL ; / / Loc al energy double s q r t d t = s q r t ( d e l t a t ) ; double D =. 5 ; / / d i f f u s i o n constant / / For Metropolis Hastings algo : double R new = ( double ) matrix (N, dimension, sizeof ( double ) ) ; double F new = ( double ) matrix (N, dimension, sizeof ( double ) ) ; double g r e e n s r a t i o ; / / Ratio between Green s f u n c t i o n s double d e t r a t i o ; / / Ratio between S l a t e r determinants double j a s t r a t i o ; / / Ratio between Jastrow f a c t o r s double rold, rnew, a ; double alphaderiv, b e t a d e r i v ;

52 52 / 95 Metropolis-Hastings part, inside Monte Carlo loop / / Ratio between S l a t e r determinants i f ( i < N2) { d e t r a t i o = 0; for ( i n t l = 0; l < N2 ; l ++) d e t r a t i o += phi ( l, R new, i, alpha ) [ l ] [ i ] ; } else { d e t r a t i o = 0; for ( i n t l = 0; l < N2 ; l ++) d e t r a t i o += phi ( l, R new, i, alpha ) DinvDown [ l ] [ i N2 ] ; } DinvUp

53 53 / 95 Metropolis-Hastings part / / Inverse S l a t e r matrix i n new p o s i t i o n i f ( i < N2) { / / Spinn up for ( i n t j = 0; j < N2 ; j ++) { i f ( j! = i ) { Sj = 0; for ( i n t l = 0; l < N2 ; l ++) { Sj += phi ( l, R new, i, alpha ) DinvUp [ l ] [ j ] ; } for ( i n t l = 0; l < N2 ; l ++) DinvUp new [ l ] [ j ] = DinvUp [ l ] [ j ] Sj DinvUp [ l ] [ i ] / d e t r a t i o ; } } for ( i n t l = 0; l < N2 ; l ++) DinvUp new [ l ] [ i ] = DinvUp [ l ] [ i ] / d e t r a t i o ; }

54 54 / 95 Metropolis-Hastings part else { / / Spinn ned for ( i n t j = 0; j < N2 ; j ++) { i f ( j! = i N2) { Sj = 0; for ( i n t l = 0; l < N2 ; l ++) { Sj += phi ( l, R new, i, alpha ) DinvDown [ l ] [ j ] ; } for ( i n t l = 0; l < N2 ; l ++) DinvDown new [ l ] [ j ] = DinvDown [ l ] [ j ] Sj DinvDown [ l ] [ i N2 ] / d e t r a t i o ; } } for ( i n t l = 0; l < N2 ; l ++) DinvDown new [ l ] [ i N2 ] = DinvDown [ l ] [ i N2 ] / d e t r a t i o ; }

55 55 / 95 Jastrow ratio / / Ratio between Jastrow f a c t o r s j a s t r a t i o = 0; for ( i n t l = 0; l < N; l ++) { i f ( l! = i ) { i f ( ( l < N2 && i < N2) ( l >= N2 && i >= N2) ) a = ; else a = 1. 0 ; r o l d = r i j (R, l, i ) ; rnew = r i j ( R new, l, i ) ; j a s t r a t i o += a ( rnew / ( 1 + beta rnew ) r o l d / ( 1 + beta r o l d ) ) ; } } j a s t r a t i o = exp ( j a s t r a t i o ) ;

56 56 / 95 Green s functions / / quantum f o r c e i n new p o s i t i o n calcqf ( R new, F new, alpha, beta,n, DinvUp new, DinvDown new, N2, detgrad new, jastgrad new ) ; / / Ratio between Green s f u n c t i o n s g r e e n s r a t i o = 0; for ( i n t i i = 0; i i < N; i i ++) for ( i n t j = 0; j < 3; j ++) g r e e n s r a t i o +=. 5 ( F new [ i i ] [ j ]+F [ i i ] [ j ] ) (. 5 D d e l t a t (F [ i i ] [ j ] F new [ i i ] [ j ] ) + R[ i i ] [ j ] R new [ i i ] [ j ] ) ; g r e e n s r a t i o = exp ( g r e e n s r a t i o ) ;

57 57 / 95 Metropolis Hastings test / / Metropolis Hastings t e s t i f ( ran2 (&idum ) < g r e e n s r a t i o d e t r a t i o d e t r a t i o j a s t r a t i o j a s t r a t i o ) { / / Accept move abd update i n v e r s S l a t e r matrix i f ( i < N2) for ( i n t l = 0; l < N2 ; l ++) for ( i n t m = 0; m < N2 ; m++) DinvUp [ l ] [m] = DinvUp new [ l ] [m] ; else for ( i n t l = 0; l < N2 ; l ++) for ( i n t m = 0; m < N2 ; m++) DinvDown [ l ] [m] = DinvDown new [ l ] [m] ;

58 / / Update p o s i t i o n, quantum f o r c e and g r a d i e n t s for ( i n t i i = 0; i i < N; i i ++) { for ( i n t j = 0; j < 3; j ++) { R[ i i ] [ j ] = R new [ i i ] [ j ] ; F [ i i ] [ j ] = F new [ i i ] [ j ] ; detgrad [ i i ] [ j ] = detgrad new [ i i ] [ j ] ; j a s t g r a d [ i i ] [ j ] = jastgrad new [ i i ] [ j ] ; } / / End loop of e l e c t r o n t h a t has been moved 58 / 95

59 59 / 95 Proof for updating algorithm of the Slater matrix As a starting point we may consider that each time a new position is suggested in the Metropolis algorithm, a row of the current Slater matrix experiences some kind of perturbation. Hence, the Slater matrix with its orbitals evaluated at the new position equals the old Slater matrix plus a perturbation matrix, where d jk (x new ) = d jk (x old ) + jk, (26) jk = δ ik [φ j (xi new ) φ j (xi old )] = δ ik ( φ) j. (27)

60 60 / 95 Proof for updating algorithm of the Slater matrix Computing the inverse of the transposed matrix we arrive to d kj (x new ) 1 = [d kj (x old ) + kj ] 1. (28) The evaluation of the right hand side (rhs) term above is carried out by applying the identity (A + B) 1 = A 1 (A + B) 1 BA 1. In compact notation it yields [D T (x new )] 1 = [D T (x old ) + T ] 1 = [D T (x old )] 1 [D T (x old ) + T ] 1 T [D T (x old )] 1 = [D T (x old )] 1 [D T (x new )] 1 T [D T (x old )] 1. {z } By Eq.28

61 61 / 95 Proof for updating algorithm of the Slater matrix Using index notation, the last result may be expanded by d 1 (x new ) = d 1 (x old ) X kj kj l = d 1 (x old ) X kj l = d 1 (x old ) X kj l X m X m X = d 1 (x old ) d 1 (x new ) kj ki = d 1 (x old ) d 1 kj m d 1 km (xnew ) T ml d 1 lj (x old ) d 1 km (xnew ) lm d 1 lj (x cur ) d 1 km (xnew ) δ im ( φ) l NX {z } By Eq. 27 d 1 lj (x old ) ( φ) l d 1 (x old ) lj l=1 NX (x new ) [φ ki l (ri new ) φ l (ri old )] l=1 {z } By Eq.27 D 1 lj (x old ).

62 62 / 95 Proof for updating algorithm of the Slater matrix Using D 1 (x old ) = adjd D(x old ) and dividing these two equations we get Therefore, d 1 kj D 1 (x old ) D 1 (x new ) = D(xnew ) D(x old ) (x new ) = d 1 kj (x old ) d 1 ki (x old ) R and D 1 (x new ) = adjd D(x new ), = R d 1 ki NX l=1 (x new ) = d 1 (x ki old ). R [φ l (r new i ) φ l (ri old )]d 1 (x old ), lj

63 63 / 95 Proof for updating algorithm of the Slater matrix or d 1 (x new ) = d 1 (x old ) kj kj + = d 1 kj (x old ) + d 1 ki (x old ) R d 1 ki (x old ) R d 1 ki (x old ) R d 1 ki (x old ) R NX l=1 NX l=1 NX l=1 NX l=1 φ l (ri new )d 1 (x old ) lj φ l (ri old )d 1 (x old ) lj d il (x new )d 1 lj (x old ) d il (x old )d 1 lj (x old ).

64 64 / 95 Proof for updating algorithm of the Slater matrix In this equation, the first line becomes zero for j = i and the second for j i. Therefore, the update of the inverse for the new Slater matrix is given by 8 >< d 1 (x d 1 (x new kj old ) d 1 (x old ) ki R ) = kj >: d 1 (x old ) ki R P N l=1 d il(x new )d 1 lj (x old ) if j i P N l=1 d il(x old )d 1 lj (x old ) if j = i

65 65 / 95 Newton s method If we consider finding the minimum of a function f using Newton s method, that is search for a zero of the gradient of a function. Near a point x i we have to second order f(ˆx) = f(ˆx i ) + (ˆx ˆx i ) f(ˆx i ) (ˆx ˆx i)â(ˆx ˆx i) giving f(ˆx) = f(ˆx i ) + Â(ˆx ˆx i). In Newton s method we set f = 0 and we can thus compute the next iteration point (here the exact result) ˆx ˆx i = Â 1 f(ˆx i ). Subtracting this equation from that of ˆx i+1 we have ˆx i+1 ˆx i = Â 1 ( f(ˆx i+1 ) f(ˆx i )).

66 66 / 95 Conjugate gradient (CG) method The success of the CG method for finding solutions of non-linear problems is based on the theory of conjugate gradients for linear systems of equations. It belongs to the class of iterative methods for solving problems from linear algebra of the type ˆx = ˆb. In the iterative process we end up with a problem like ˆr = ˆb ˆx, where ˆr is the so-called residual or error in the iterative process.

67 67 / 95 Conjugate gradient method The residual is zero when we reach the minimum of the quadratic equation P(ˆx) = 1 2ˆxT ˆx ˆx T ˆb, with the constraint that the matrix  is positive definite and symmetric. If we search for a minimum of the quantum mechanical variance, then the matrix Â, which is called the Hessian, is given by the second-derivative of the variance. This quantity is always positive definite. If we vary the energy, the Hessian may not always be positive definite.

68 68 / 95 Conjugate gradient method In the CG method we define so-called conjugate directions and two vectors ŝ and ˆt are said to be conjugate if ŝ T ˆt = 0. The philosophy of the CG method is to perform searches in various conjugate directions of our vectors ˆx i obeying the above criterion, namely ˆx T i ˆx j = 0. Two vectors are conjugate if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if ŝ is conjugate to ˆt, then ˆt is conjugate to ŝ.

69 69 / 95 Conjugate gradient method An example is given by the eigenvectors of the matrix ˆv T i ˆv j = λˆv T i ˆv j, which is zero unless i = j.

70 70 / 95 Conjugate gradient method Assume now that we have a symmetric positive-definite matrix  of size n n. At each iteration i + 1 we obtain the conjugate direction of a vector ˆx i+1 = ˆx i + α iˆp i. We assume that ˆp i is a sequence of n mutually conjugate directions. Then the ˆp i form a basis of R n and we can expand the solution ˆx = ˆb in this basis, namely ˆx = nx α iˆp i. i=1

71 71 / 95 Conjugate gradient method The coefficients are given by nx Ax = α i Ap i = b. i=1 Multiplying with ˆp T k from the left gives nx ˆp T k ˆx = α iˆp T k ˆp i = ˆp T ˆb, k i=1 and we can define the coefficients α k as α k = ˆpT ˆb k ˆp T k ˆp k

72 72 / 95 Conjugate gradient method and iterations If we choose the conjugate vectors ˆp k carefully, then we may not need all of them to obtain a good approximation to the solution ˆx. So, we want to regard the conjugate gradient method as an iterative method. This also allows us to solve systems where n is so large that the direct method would take too much time. We denote the initial guess for ˆx as ˆx 0. We can assume without loss of generality that ˆx 0 = 0, or consider the system Âẑ = ˆb ˆx 0, instead.

73 73 / 95 Conjugate gradient method Important, one can show that the solution ˆx is also the unique minimizer of the quadratic form f(ˆx) = 1 2ˆxT ˆx ˆx Tˆx, ˆx R n. This suggests taking the first basis vector ˆp 1 to be the gradient of f at ˆx = ˆx 0, which equals ˆx 0 ˆb, and ˆx 0 = 0 it is equal ˆb. The other vectors in the basis will be conjugate to the gradient, hence the name conjugate gradient method.

74 74 / 95 Conjugate gradient method Let ˆr k be the residual at the k-th step: ˆr k = ˆb ˆx k. Note that ˆr k is the negative gradient of f at ˆx = ˆx k, so the gradient descent method would be to move in the direction ˆr k. Here, we insist that the directions ˆp k are conjugate to each other, so we take the direction closest to the gradient ˆr k under the conjugacy constraint. This gives the following expression ˆp k+1 = ˆr k ˆpT k ˆr k ˆp T k ˆp ˆp k. k

75 75 / 95 Conjugate gradient method We can also compute the residual iteratively as ˆr k+1 = ˆb ˆx k+1, which equals or which gives ˆb Â(ˆx k + α k ˆp k ), (ˆb ˆx k) α k ˆp k, ˆr k+1 = ˆr k ˆp k,

76 76 / 95 Codes from numerical recipes The codes are taken from chapter 10.7 of Numerical recipes. We use the functions dfpmin and lnsrch. You can load down the package of programs from the webpage of the course, see under project 1. The package is called NRcgm107.tar.gz and contains the files dfmin.c, lnsrch.c, nrutil.c and nrutil.h. These codes are written in C. void dfpmin(double p[], int n, double gtol, int *iter, double *fret, double(*func)(double []), void (*dfunc)(double [], double []))

77 77 / 95 What you have to provide The input to dfpmin void dfpmin(double p[], int n, double gtol, int *iter, double *fret, double(*func)(double []), void (*dfunc)(double [], double [])) is The starting vector p of length n The function func on which minimization is done The function dfunc where the gradient i calculated The convergence requirement for zeroing the gradient gtol. It returns in p the location of the minimum, the number of iterations and the minimum value of the function under study fret.

78 78 / 95 Simple example and demonstration For the harmonic oscillator in one-dimension with a trial wave function and probability ψ T (x) = e α2 x 2, P T (x)dx = e 2α2 x 2 dx R dxe 2α 2 x 2 with α as the variational parameter. We have the following local energy E L [α] = α 2 + x α2 «, which results in the expectation value E L [α] = 1 2 α α 2

79 79 / 95 Simple example and demonstration The derivative of the energy with respect to α gives d E L [α] dα = α 1 4α 3 and a second derivative which is always positive (meaning that we find a minimum) d 2 E L [α] dα 2 = α 4 The condition gives the optimal α = 1/ 2. d E L [α] dα = 0,

80 80 / 95 Simple example and demonstration In general we end up computing the expectation value of the energy in terms of some parameters α = {α 0, α 1,..., α n}) and we search for a minimum in parameter space. This leads to an energy minimization problem. The elements of the gradient are (Ei is the first derivative wrt to the variational parameter α i ) fi ψi Ē i = ψ E L + Hψ i ψ 2Ē ψ fl i (29) ψ fi fl ψi = 2 ψ (E L Ē) (by Hermiticity). (30) For our simple model we get the same expression for the first derivative (check it!).

81 81 / 95 Simple example and demonstration Taking the second derivative the Hessian is " fi ψij Ē ij = 2 ψ + ψ «fl iψ j ψ 2 (E L Ē) fi fl fi fl fi fl # ψi ψj ψi Ē j Ē i + ψ ψ ψ E L,j. (31) Note that our conjugate gradient approach does need the Hessian! Check again that the simple models gives the same second derivative with the above expression.

82 82 / 95 Simple example and demonstration We can also minimize the variance. In our simple model the variance is with first derivative σ 2 [α] = 1 2 α α 4, dσ 2 [α] dα = 2α3 1 8α 5 and a second derivative which is always positive d 2 σ 2 [α] dα 2 = 6α α 6

83 83 / 95 Conjugate gradient method, our case In Newton s method we set f = 0 and we can thus compute the next iteration point (here the exact result) ˆx ˆx i = Â 1 f(ˆx i ). Subtracting this equation from that of ˆx i+1 we have ˆx i+1 ˆx i = Â 1 ( f(ˆx i+1 ) f(ˆx i )).

84 84 / 95 Simple example and demonstration In our case f can be either the energy or the variance. If we choose the energy then we have ˆα i+1 ˆα i =  1 ( E(ˆα i+1 ) E(ˆα i )). In the simple model gradient and the Hessian  are d E L [α] dα = α 1 4α 3 and a second derivative which is always positive (meaning that we find a minimum)  = d2 E L [α] dα 2 = α 4

85 85 / 95 Simple example and demonstration We get then α i+1 = 4 3 α i α4 i 3α 3, i+1 which can be rewritten as α 4 i α iα 4 i α4 i. Our code does however not need the value of the Hessian since it produces an estimate of the Hessian.

86 86 / 95 Simple example and code (model.cpp on webpage) #include "nrutil.h" using namespace std; // Here we define various functions called by the main program double E_function(double *x); void de_function(double *x, double *g); void dfpmin(double p[], int n, double gtol, int *iter, double *fret, double(*func)(double []), void (*dfunc)(double [], double [])); // Main function begins here int main() { int n, iter; double gtol, fret; double alpha; n = 1; cout << "Read in guess for alpha" << endl; cin >> alpha;

87 87 / 95 Simple example and code (model.cpp on webpage) // reserve space in memory for vectors containing the variational // parameters double *p = new double [2]; gtol = 1.0e-5; // now call dfmin and compute the minimum p[1] = alpha; dfpmin(p, n, gtol, &iter, &fret,&e_function,&de_function); cout << "Value of energy minimum = " << fret << endl; cout << "Number of iterations = " << iter << endl; cout << "Value of alpha at minimu = " << p[1] << endl; delete [] p;

88 88 / 95 Simple example and code (model.cpp on webpage) // this function defines the Energy function double E_function(double x[]) { double value = x[1]*x[1]* /(8*x[1]*x[1]); return value; } // end of function to evaluate

89 89 / 95 Simple example and code (model.cpp on webpage) // this function defines the derivative of the energy void de_function(double x[], double g[]) { g[1] = x[1]-1.0/(4*x[1]*x[1]*x[1]); } // end of function to evaluate

90 90 / 95 Using the conjugate gradient method Start your program with calling the CGM method (function dfpmin). This function needs the function for the expectation value of the local energy and the derivative of the local energy. Change the functions func and dfunc in the codes below. Your function func is now the Metropolis part with a call to the local energy function. For every call to the function func I used 1000 Monte Carlo cycles for the trial wave function Ψ T (r 1, r 2 ) = e α(r 1+r 2 ) This gave me an expectation value for the energy which is returned by the function func. When I call the local energy I also compute the first derivative of the expectaction value of the local energy d E L [α] dα fi fl ψi = 2 ψ (E L[α] E L [α] ).

91 91 / 95 Using the conjugate gradient method The expectation value for the local energy of the Helium atom with a simple Slater determinant is given by E L = α 2 2α Z 5 «16 You should test your numerical derivative with the derivative of the last expression, that is d E L [α] = 2α 2 Z 5 «. dα 16

92 92 / 95 Simple example and code (model.cpp on webpage) #include "nrutil.h" using namespace std; // Here we define various functions called by the main program double E_function(double *x); void de_function(double *x, double *g); void dfpmin(double p[], int n, double gtol, int *iter, double *fret, double(*func)(double []), void (*dfunc)(double [], double [])); // Main function begins here int main() { int n, iter; double gtol, fret; double alpha; n = 1; cout << "Read in guess for alpha" << endl; cin >> alpha;

93 93 / 95 Simple example and code (model.cpp on webpage) // reserve space in memory for vectors containing the variational // parameters double *p = new double [2]; gtol = 1.0e-5; // now call dfmin and compute the minimum p[1] = alpha; dfpmin(p, n, gtol, &iter, &fret,&e_function,&de_function); cout << "Value of energy minimum = " << fret << endl; cout << "Number of iterations = " << iter << endl; cout << "Value of alpha at minimu = " << p[1] << endl; delete [] p;

94 94 / 95 Simple example and code (model.cpp on webpage) // this function defines the Energy function double E_function(double x[]) { // Change here by calling your Metropolis function which // returns the local energy double value = x[1]*x[1]* /(8*x[1]*x[1]); return value; } // end of function to evaluate You need to change this function so that you call the local energy for your system. I used 1000 cycles per call to get a new value of E L [α].

95 95 / 95 Simple example and code (model.cpp on webpage) // this function defines the derivative of the energy void de_function(double x[], double g[]) { // Change here by calling your Metropolis function. // I compute both the local energy and its derivative for every call g[1] = x[1]-1.0/(4*x[1]*x[1]*x[1]); } // end of function to evaluate You need to change this function so that you call the local energy for your system. I used 1000 cycles per call to get a new value of E L [α]. When I compute the local energy I also compute its derivative. After roughly iterations I got a converged result in terms of α.