So, today we will. So, I am going to outline this topic on jointly Gaussian random variables. So, because I have only about an R and this is a very important and fairly detailed topic. So, I will skip proofs and such I will only give you an introduction and I will and you can look up some material if you want more details. So, you know the one dimensional Gaussian. So, the p d f looks like this right looks like 1 over sigma square root of 2 pi e power minus x square by well x minus mu square minus mu square over 2 sigma square right. So, this is a Gaussian with mean mu and variance sigma square. So, now. So, we want to discuss today jointly Gaussian random variables or random Gaussian random vectors. So, we want to talk about random vector x 1 through x n. So, this is nothing but a these are all random variables and we want to talk about what it means to what it means for a vector of random variables to be a Gaussian random vector or jointly Gaussian random variables x 1 through x n. See one thing in particular I want to say right away is that a random vector is not said to be a Gaussian random vector if each of the entries is Gaussian.  It is not the case that this is considered a Gaussian random vector a Gaussian random vector is much more than each of the entries being marginally Gaussian. So, in a jointly Gaussian vector in a Gaussian vector or jointly Gaussian random variables each of these guys will have a marginal which is Gaussian, but that is not sufficient it is much more than that that is what we are going to talk about today. So, in two dimensions. So, bivariate. So, let us first talk about a bivariate Gaussian and then generalize to a multivariate Gaussian or a Gaussian random vector. So, you say that x and y are bivariate Gaussian. So, x and y are some random variables which live in the same probability space omega f p if the joint density f x comma y looks like this. So, it will be a standard bivariate. Standard bivariate Gaussian if f x y of x y looks like 1 over 2 pi square root of 1 minus rho squared e power minus x square minus 2 rho x y plus y square over twice 1 minus rho square. So, this is the standard bivariate Gaussian density here rho is a parameter that lies in minus 1 1. So, in this. So, what you can show is this I will not show this proposition. So, first of all you have to verify that this is indeed a valid pdf by showing the integrates to 1. So, the standard trick of doing that is to complete squares the top this is standard trick to do that. So, this is indeed a valid pdf in two dimensions which is easy to show when property a x and y are marginally distributed as n 0 1 b rho of x y the correlation coefficients the correlation coefficient between x and y is equal to rho and c. The conditional density of x given y is. So, this is normal with mean rho y and variance 1 minus rho square. So, we say that this x and y is a bivariate Gaussian if this is your joint pdf. So, what we can show is if you integrate one variable out let us say integrate y out you will get a standard Gaussian in x and vice versa if you integrate with symmetric in x and y. So, if integrate x out you will get a standard Gaussian in y. So, indeed x and y are marginally distributed as n 0 1. So, x and y are standard Gaussians marginally, but they are not independent. So, we are saying that. So, if they are independent the density will simply look like the product of the two Gaussian densities. In this case it is a more complicated form in fact, you can show that this number rho is in fact, the correlation coefficient between x and y and the way you do that is to explicitly compute expectation of x y put x y and integrate this out you will get rho which is the correlation coefficient between these the standard Gaussians. And the conditional density of x given y this you can compute using just dividing this by the density the marginal density of y which is a standard Gaussian. From there you can conclude that the conditional density is also Gaussian with mean rho y and variance 1 minus rho square these you can directly compute from this expression. So, that is a very simple. So, this is the standard bivariate Gaussian. So, one remarkable fact about this bivariate Gaussian or in general actually all multivariate Gaussians this is true is that if you have uncorrelated random variables x and y are uncorrelated then you know that rho is 0. Correlation coefficient will be 0. So, you put rho equal to 0 here what happens you get 1 over 2 pi e to the minus x square plus y square over 2 right which looks like the product of the marginals. So, the moment you put rho equal to 0 it immediately implies that yes it immediately implies that the random variables are independent correct. So, normally we know that independence implies the random variables are uncorrelated, but the converse is certainly not true uncorrelated random variables are not necessarily independent. However, in the jointly Gaussian world. So, if x and y are jointly Gaussian or which is the same as saying the bivariate Gaussian then uncorrelated jointly Gaussian random variables are independent. So, that is one remarkable property that comes out right from here. Another interesting thing that you can look at this and figure out is that. So, if you want to compute something like let us if you want to compute the conditional expectation of x given y. So, from here you can compute the conditional expectation of x given y because conditional expectation of x given y equal to little y is equal to rho little y right because. So, this is your conditional density. So, from here you can easily figure out that expectation of x given y equal to little y is nothing, but I mean of this Gaussian which is rho times little y right. So, this conditional expectation will be what rho times big y correct. So, this implies that if you have if all this x. So, if x and y are jointly Gaussian or bivariate Gaussian then expectation of x given y is a linear function of y. So, in the signal processing people say that. So, this is the MMSE estimate right and the MMSE estimate is a linear function right. So, linear estimate. So, linear estimate is optimal in the jointly Gaussian world. This is something that signal processing people use a lot if x and y are jointly Gaussian the MMSE estimate is a linear function of your observation. Why is what you observe right. So, that is just some this is in the this is just two dimensions. So, if you look at this density itself. So, you have to plot this. So, you have to plot this x y and you have to plot the density right. So, it will be some three dimensional curve right. So, if your rho is 0 if they are independent then you will have a nice symmetric bell curve in all directions right. You will have a nice you just plot it in matlab it is very nice to see right. It will be symmetric in all directions no matter which way you cut it will look like a n 0 1. But if your rho is not 0 the distribution is a little skewed in one way right. So, what. So, if you plot this distribution right for some rho value which is not equal to 0 and then you let us say you slice the pdf and you plot its level sets as they are called right. Let us say if this is your pdf in three dimensions you slice it at various levels and you plot it plot the level sets in that case you will get. So, the level sets will look like this. So, they are all 0 mean and so if rho is 0 of course you will get these concentric circles right which means like the Gaussian is symmetric in all directions independent case. If your rho is positive you get something that you get these concentric ellipses centered at the origin. So, I have drawn it poorly. So, this is centered at the origin and you get these concentric ellipses. So, the Gaussian will look kind of sharp in this axis and quite spread out in that axis. So, the variance along this direction and this direction will both be equal to 1. So, this is for the case rho greater than 0 and for so which means rho is greater than 0 means that if x is positive y is more likely to be positive. That is why the ellipses are like this right there is more mass along here than along here. There is more mass along first and third quadrants than second and four quadrants. Similarly, if you have rho less than 0 you will have ellipses that look like that. You just plot it in matlab and see the best way to see this is to actually generate this contour well the surface plots in matlab. So, more generally if you want to talk about a non standard bivariate Gaussian the density will be even more messy. So, this is already messy even for a 0 mean univariate case. So, I do not even know I should put if I should put it down it is a fairly messy expression here I go let me put it down.  Close bracket and close that bracket 2 rho I think I got that right. So, it is a big mess. So, it is a big mess to write it down explicitly. So, this guy has mean of mu 1 comma mu 2 for x and y. So, expectation of x will be mu 1 expectation of y will be mu 2 and rho as usual will be the correlation coefficient and sigma 1 and sigma 2 the variance of variances of x x 1 y respectively. So, thankfully if you use a matrix notation this becomes easier to notate. So, the same thing use the matrix notation. So, this becomes like this. So, this whole guy this whole expression can be written as 1 over I think it is a 2 pi missing somewhere. So, this should be 2 pi here 1 by 2 pi determinant of v e power minus x. So, vector x minus vector mu v inverse x minus mu transpose over 2 where. So, actually let me call this f x 1 x 2 x 1 x 2 of x 1 comma x 2. So, wherever this y let me write x 2. So, here. So, my vector x is x 1 x 2 and my vector mu is mu 1 mu 2 this is square root here and v is sigma 1 squared sigma 2 squared. So, basically this will be of the form this will be like. So, this will be variance of x 1 variance of x 2 and this will have covariance of x 1 x 2 and covariance of x 1 x 2 you can write in terms of rho and sigma 1 and sigma 2. You can actually you can just explicitly see that these 2 are equal. So, this is the mean vector the vector of means expectation of x 1 expectation of x 2 and this is called the covariance matrix. So, it is nothing but the expectation of. So, x is a rho vector is not it. So, it is the expectation of x x minus mu. So, x minus mu x minus mu transpose which means you will write all the variances of the diagonals and covariance of x 1 x 2 1 on the off diagonals. So, generally what determines. So, let us just. So, if you just look at a more general joint Gaussian the center of these ellipses will be shifted to mu 1 comma mu 2 and your variances will decide the spread of these ellipses in both axis and what you can also show is that this these 2 axis the major axis and minor axis of the ellipse are. In fact, the Eigen vectors of are determined by the Eigen vectors of this covariance matrix. See this covariance matrix can be easily shown to be a positive semi definite positive definite in case it is actually non singular. So, if for a positive definite matrix the Eigen values defined an orthonormal basis in the space right. So, this these 2 orthonormal vectors define the axis of the ellipse. So, the where this is oriented this determined by the Eigen vectors of this matrix. So, basically the multivariate Gaussian in the n dimensional case is defined exactly similarly. What you do is now it is exactly the same formula except now you will have a vector of n random variables and mean vector will be n long and the covariance matrix will have variance x 1 variance x 2 variance x n on the diagonals and all the correlation terms on of that of the covariance terms of the diagonals. Oh by the way. So, before I go to the n dimensional case I want to. So, I made a remark in the beginning that x x n y x 1 and x 2 being both marginally Gaussian does not mean that x 1 x 2 is a bivariate Gaussian right. So, I just want to give a explicit example show that that is not the case. Let us say here example let y 1 y 2 be iid random variables distributed according to the pdf f y of y equal to square root of 2 over pi e power minus y square over 2 for y greater than or equal to 0 and 0 otherwise. So, this is how this is a pdf this you can actually clearly you can you can integrate this and see that this is a pdf and y 1 and y 2 are independent and identically distributed according to this pdf. So, if you look at this pdf what is this look like? It is like Gaussian with one side chopped off it looks like f y looks like this. So, it is as though the negative side of the Gaussian has been chopped off and all its mass has been put on the positive side. So, actually what you are doing is taking a Gaussian and taking its absolute value that is what this random variables. So, this is f y of little y versus y right late. So, these 2 are iid w equal to plus 1 with probability half minus 1 with probability half be independent of y 1 and y 2 define x 1 equal to w y 1 and x 2 equal to w y 2. So, I am constructing an example here. So, what will be the marginal density of x 1? So, for x 1 I am generating y 1 according to this distribution and then tossing a coin which will either come up heads or tails plus 1 or minus 1 with probability half right. So, my x 1 will be equal to the value of y 1 with probability half or minus 1 minus y 1 with probability half correct. So, with. So, my density of x 1 will have a density. So, you can show that the density of x 1 is with probability half it will have a density like that with probability half will have a density like that right. So, you can show I think this is right I think I am not making any mistakes tell me if I do x 1 is n 0 1 and similarly x 2 is also n 0 1 same argument. So, both x 1 and x 2 are marginally standard Gaussian random variables right they marginally both Gaussian, but the contention is that, but x 1 x 2 are not is not a bivariate Gaussian right. In other words they are not jointly Gaussian is not a bivariate Gaussian why is that x 1. So, I am looking at the vector x 1 and x 2 right when. So, in order to show that something is a bivariate Gaussian I need to show that it has a density of that form right minus 2 rho x y 1 naught right, but clearly x 1 and x 2 cannot possibly have a density of that bivariate form bivariate Gaussian form. So, if you notice so the same w is feeding to the signs of x 1 and x 2 see y 1 and y 2 are always positive correct always non negative and w decides the sign of these x's, but the same w feeds in here right. So, if my w is plus 1 here it is also going to be plus 1 here which means if I look at my x 1 and realize that it is positive then I can immediately say that x 2 is also positive there is no way that x 1 and x 2 will have different signs in other. So, which means that if you just plot little x 1 and little x 2 the joint density of x 1 and x 2 is constrained to the first and third quadrants correct. So, looking at the so x 1 alone is a standard Gaussian x 2 alone is a standard Gaussian, but the joint density is only constrained to the first quadrant and third quadrant by construction the density is 0 here and here agreed in particular x 1 and x 2 are certainly not independent right, but and the density is only constrained to the first and third quadrant which means you do not have a situation like this. If it were jointly Gaussian then you will have contours like this right I mean there is non 0 mass everywhere no matter what your rho is you have non 0 mass everywhere right. So, this cannot possibly be a bivariate Gaussian then you have so then so that is a case of a degenerate that is a degenerate case that is a degenerate case that is the case of rho equal to 1 actually.  So, we will talk about joint Gaussian since n dimensions also where you will get situations where in n dimensions the density may exist in a subspace of k dimensions. So, in that case you can just treat it as a k dimensional Gaussian. So, in this case yeah so that is what you do normally. So, is this clear this example clear this is very instructive because it says that bivariate Gaussian is saying more than saying that they are marginally Gaussian. So, in the n dimensional multivariate case so we will define for n dimensions now. So, there are three equivalent definitions of a multivariate Gaussian or a joint Gaussian. So, my x is so column vector of n random variables x is said to be multivariate Gaussian or jointly Gaussian x 1, x n are said to be jointly Gaussian if it has pdf f x of x equal to 1 over square root of 2 pi to the n determinant of v e power minus x minus mu transpose v inverse of x minus mu. So, this is a multivariate case x minus mu divided by 2 where mu is a real vector and v is a positive definite matrix. So, we are just putting down the same formula in n dimensions and of course. So, I have not it in this. So, from here so this is a pdf you can show this is a valid pdf by making some appropriate substitutions and integrating it out. What we can show is that mu is the vector of expectations expected x 1 expected x 2 and so on and v is the covariance matrix. All the diagonal elements will be the variances and the off diagonal elements will be the covariance expected x i x j terms covariance x i x j terms. So, this is one definition with explicitly specifies the pdf. So, if you want to check if some vector is a joint Gaussian vector or multivariate Gaussian vector you go and see if it has this form for some positive definite v and some vector mu. If it does not it is not if it does it is. So, this is the most explicit specification of a multivariate Gaussian. However, this is very messy to work with even in 2 dimensions if you write this out it was a tremendous right. So, this is not very I mean it leads to T d s integrals and T d s computations right. So, often there are there are equivalent definitions there are actually 2 more equivalent definitions which are commonly used which I will also put down. But, the thing is if I put down multiple definitions you have to show that they are all equivalent right. So, I am going to put down 3 definitions. So, I have to show how many implications 6 implications altogether right. So, I will not do that, but I am saying that you can do it. Definition 2 first line is the same x is said to be a Gaussian vector if it can be expressed as x is equal to w plus mu where w is a vector of i i d n 0 1 random variables. So, this is an equivalent definition you say that x is a multivariate Gaussian if it can be obtained by a affine transformation of a i i d Gaussian vector. So, this w is something you can perfectly understand each of the entries is standard Gaussian and they are all independent. So, w 1 w 2 w n they are all i i d n 0 1 random variables you take that vector and perform this affine transformation on it d is some vector. So, d is a matrix. So, for some matrix d. So, d is a matrix. So, it is some affine transformation of mu is a real vector mu is in R n. So, you can show that I mean you can show the equivalence of these two definitions.  And definition 3. So, definition 3 is probably the most cryptic, but probably the most elegant also. So, x is said to be a multivariate Gaussian if for every a vector a in R n a transpose x is a Gaussian random variable. So, this definition says that x is said to be a Gaussian random vector or a multivariate Gaussian if every linear combination of the x size.  So, on a n x n if every linear combination of these x is in fact Gaussian distributed. So, even if you find some vector a for which is not Gaussian distributed then x is not a multivariate Gaussian. So, may be perhaps you can find take it as an exercise in this case try to find some a 1 and a 2 for which a 1 x 1 plus a 2 x 2 does not have a Gaussian distribution. It is fairly easy I think because they have the same sign. So, you just have to figure out some a 1 and a 2 for which a 1 x 2 x 1 plus a 2 x 2 is not Gaussian distributed. So, I mean in particular. So, you can take a equal to 0 and then you will get 0. So, you have to basically accept the 0 random variable as a very special case of a Gaussian with 0 mean and 0 variance. So, subject to that understanding this definition is fine great. So, these three definitions are in fact equivalent. It can be proven actually the proof is available in. So, MIT course where I think.  So, the class i has a 0 in which case they aims to be Gaussian. So, this definition clearly says that is margins are marginally Gaussian, but it says more than that, any other questions? So, these three definitions are equivalent. So, another just like in a two dimensional case, you can show that if the. So, this covariance matrix v as I said contains all the increases in a sugar variances and the co variances on or the off diagonals. So, this is a positive definite matrix in general and it is. So, it is Eigen vectors which define a orthonormal basis for the space will again tell you the in which directions the Gaussian will spread out more on which direction the Gaussian is not is more compact in more is more compressed alright. So, this is in n dimensions. So, we cannot draw it, but just the two dimensional analogy can be pushed forward. Also you can show that if the diagonal if the matrix V has a diagonal structure which means or the co variances are 0 then what you can show is that this will actually product out into. So, V inverse will simply be. So, it will be like the sigma 1 to the minus 2, sigma 2 to the minus 2 and so on. So, if you expand this out you will actually get a product of the marginal p d f's. So, which again means that if you have a diagonal matrix for your co variance matrix that necessarily implies that the random variables are independent right. So, generally uncorrelated entries the entries in the Gaussian vector are uncorrelated then they are all independent which is generally not true only for Gaussian's vector is it true. So, even in this case I mean there are some results about see you can just like in the two dimensional case you can show that the conditional expectation is a linear function. So, which means the MMSE estimate is a linear function of your observation. So, linear filtering is optimal is what these signal processing people like to say right. So, linear filtering is optimal in the sense of MMSE for jointly Gaussian random variables. So, there is a couple of other cool facts may be I should put down about these jointly Gaussian's before I stop it as you see I am not spending too much detail on anything I am just pointing out some interesting facts. So, if you have x is equal to x 1 dot x n let say this is a multivariate let say this is a 0 mean multivariate Gaussian with co variance matrix V. So, I am just throwing out the means because means are just a headache right you can if they have if they have non negative non 0 means just subtract it out subtract the mean vector out. So, in this case you can perform. So, these guys are. So, this x is. So, the idea is to in some sense generate. So, this x is given to you this is a multivariate Gaussian vector with some co variance matrix. Now, we are going to generate uncorrelated or independent Gaussian random variables from here. So, this so the procedure is to do this you take x 1 w 1 equal to x 1 then you take w 2 equal to. So, you take x 2 and subtract out that. So, you know that this error. So, this is the estimation error in estimating x 2 looking at just looking at x 1 you know that this is orthogonal to that guy right. But, they are Gaussian the joint Gaussian therefore, they are actually independent not just uncorrelated they are actually independent right and if you want to normalize it again you can divided by it is variance or you know you can make it unit variance if you like. But, they can proceed like this right w 3 I can write as x 3 minus conditional expectation of x 3 given x 1 x 2. So, I so here again I am projecting. So, I am estimating x 3 based on observing x 1 and x 2 and again this error will be orthogonal to the subspace span by x 1 and x 2 correct and so on right I can keep doing this. So, this is called this is known variously as Gram-Schmidt procedure in some people call this Gram-Schmidt procedure signal processing people call it whitening ok. So, it is called this is called whitening because. So, I mean I guess this is from the noise literature right. So, you have this. So, you have this colored variable. So, to speak it as all these correlations and then you have these w's are uncorrelated and you can you can make it unit variance if you like. So, it is like in all these dimensions it has equal components. So, it is called whitening of this x i's this procedure is called whitening and it is a causal procedure right. So, I look at x 1 and x 2 and then decide my next w 3 and so on right. And since see one thing you should know is that if you look at this guy this will be a linear function of x 1 and x 2 because conditional expectations are linear in whatever you observe correct. So, this whole operation can be written as a vector w equal to l times x where l is a lower triangular matrix. So, you see what I mean. So, this will only be a this will be a linear function of x 1 this will be a linear function of x 1 and x 2 and so on. So, you have only a set of linear operations and so w. So, you can write this as a matrix form w equal to l x and because of this structure you get a lower triangular matrix this is called whitening matrix. So, you can whiten this vectors whiten this vector and you can in fact show that covariance of the covariance matrix right covariance of x x transpose or whatever. So, this was v is not it can be written as l inverse whitening filter times b times l inverse transpose or which is the same as where b is a diagonal matrix. So, you can write this as l inverse b to the half b to the half l inverse transpose. So, this will also be a lower triangular matrix this will be an upper triangular matrix and. So, this is like a triangular square root of your covariance matrix. So, this is called Koleski factorization Koleski decomposition anyway. So, that is about whitening which is again common in signal processing. Finally, the one another. So, I will stop with one more little nugget about this multivariate Gaussian. So, you know that if you have just let us say any let us say single variate Gaussian you know that the nth moment of the single variate Gaussian is only a function of mu and sigma right it is completely specified by mu and sigma square. Similarly, if you have a Gaussian vector x 1 through x n all moments like expectation of x 1 to the 4 x 2 to the 13 whatever any moment of any order can be figured out by just looking at the mean vector and covariance matrix. So, the theorem that says this is called Wicks theorem. So, let us say that let us say x 1 x 2 dot x n is a 0 mean let us throughout the mean because it is just a headache 0 mean multivariate Gaussian. Then expectation of let us say I am going to write little y 1 little maybe I should write maybe I should write expectation of y 1 y 2 y n equal to 0 for n odd and expectation of y 1 y 2 dot y 2 n. So, I am looking at an even product that is given by some pi expectation of y i y j see this y. So, the reason I switched from x is to y I did not do this as a mistake each of these y's could be any of the x's. So, in y 1 you can put x 13 y 2 can again be x 13 if you like y 3 can be x 1 right you can put it in any way you want see these y's are some x's. So, if you have an odd product because they are 0 mean you will have 0 and for the even case you will have all possible 2 way products of these covariance terms. So, this denotes all possible sums of these y i's taken to a written. So, for example, if you want to compute. So, for example, if you want to compute expectation of let us say if you want to compute expectation of y 1 y 2 y 3 y 4 even with repetitions y 1 could be x 1 y 3 could also be x 1 even with repetition. So, this will simply be expectation of y 1 y 2 expectation of y 3 y 4 plus expectation of y 1 y 3 expectation of y 2 y 4 plus expectation of y 1 y 4 times expectation of y 2 y 3 is that all. So, I have to take all I have 4 terms I have to take all possible pairs and then add this all and similarly if I have even for longer expressions you can do it even if you have x 1 to the x 1 squared x 2 to the 4th some x 3 x 3 to the 6th you will have to expand the whole thing out in terms of the covariance these are all covariance matrix entries and each of these y's are some x i's. So, what this Wicks theorem says is if you are given the covariance matrix namely these entries you can go ahead and compute any moment of these joint Gaussians ok. So, this is called Wicks theorem sometimes it is also called Feynman diagram formula because Feynman used it in some calculation in his quantum field theory work he did not invent it right it was much older than him, but he used it. So, it is called Feynman diagram formula. So, that is another nice negate about this joint Gaussians yes. So, that was my crash course on joint Gaussians random variables. This n ordered n even yes can we interpret with geometry on the y it is 0 on y this is some. So, no it is very simple. So, you have y 1 y 2 blah blah right. So, even if these are repeated let us say I have y 1 y 2 y 3. So, there is no way for me to take. So, even if I have y 1 y 2 y 3. So, y 1 and y 2 could be the same or different, but one of them will always have to be different. So, the expectation will be 0, but even case this may be like y 1 square y 2 square x 1 square x 2 square in which case I have to write all the x 1 x 1 x 2 x 2 x 1 x 2 at all these possibilities I have to exhaust right. So, this is what I am going to do. So, this is what I am going to do. So, I am going


Let me go ahead little bit around and discuss something more on Hartree-Fock, so some more on Hartree-Fock. So, let us see what more we want to do. And we have particularly discussed RHF, I want to again tell you. We have particularly discussed restricted Hartree-Fock determinant and we have mentioned that this is a singlet state. What is a singlet state or a triplet state means it is an eigenfunction of S square and one of the S operator conveniently it is taken as Sz just like angular momentum. So you have for eigenfunction of sorry, S square, for eigenfunction of S square you have square root S into S plus 1 over h cross and for Sz you have just MS h cross, so these are the two eigenvalues that you get and when you do this S square the capital S or S that comes in the S into S plus 1 is 0. So that is why it is called singlet, okay. Now there is an important theorem for a determinant I might have already mentioned but again I should mention this, when is a determinant spin adopted? So this is called spin adopted, so what is a spin adopted state? The spin adopted determinant or a state is an eigenfunction of S square and is one of the S operators, so that is conveniently Sz operator, okay. So now what we want to say ask the question when is a determinant spin adopted? I hope the question is clear, when is it an eigenfunction of S square and Sz operator? Again this proof is very elaborate but the result is very important. I want to make the result very clear. So when you discuss spin adaptation remember we have to worry about what kind of spin it is, so how many of them are alpha spin, how many of them are beta spins, so that is very important. So obviously we have to write not in terms of general spin orbitals but a space part and alpha and beta attached. So let us assume that in a determinant we have n alpha spin orbitals and this is a very general statement that we are making n alpha spin orbitals of alpha spin. I hope the point is very clear okay and n beta number of spin orbitals which are beta spin such that of course n is equal to n alpha plus n beta that is very clear because you have only two spins. So you have n spin orbitals, I am categorizing the n spin orbitals as sum with alpha spin sum with beta spin okay that is what is that is the only possibility that you have right. And let us say that n alpha number is alpha spin, n beta number is beta spin such that the total n is n alpha plus n beta is it clear? So this is a very general statement I am making in a determinant or a single determinant when I have got chi 1, chi 2 to chi n if I analyze them for RHS of course it is very clear that n alpha is equal to n beta correct and each of them is n by 2 half of n so it n is equal to n alpha plus n beta. So both the number two numbers are identical for RHS but in general I can have n alpha number of alpha spin and n beta number of beta spin then there is a theorem which states which is the following theorem then let us assume one of them is maybe larger than the other so n alpha is greater than equal to n beta let us say I mean it could be the other way round of course equality equal to beta is also assumed so it is either greater or equal to n beta then the theorem says that the determinant will be spin adopted again I told you what is spin adopted. Spin adopted means it is an eigen function of s square and s z so that is spin adaptation okay for singlet the value of s is 0 but n is triplet is also spin adopted so determinant will be spin adopted if and only if this is very important if and only if the lower number so let us say here n beta number of space orbitals or n beta space orbitals with beta spin which means those spin or space orbitals which have beta spin clearly those numbers are n beta or subset or fully subsumed of subsumed in or of whatever of the n alpha space orbitals with alpha spin which means the lower number of spin orbitals have this space part these space parts are complete subset of the larger n alpha space parts I hope that point is clear now what you mean by subset I hope you understand so for example let me give a given example so 1s alpha a determinant not necessarily R H F 1s alpha sorry 2s alpha 1s beta 3 electron does this follow this sorry 2s alpha yeah does this follow this yes because your number of space orbital for beta part is 1s which is completely contained here for alpha you have 1s 2s you have 1s so this ticks this box so this is a subset of this that means these number of space orbitals have to be already contained in the n alpha space orbitals least so they are common in addition of course n alpha will have more because n alpha is greater than equal to n beta in which case determinant will be spin adopted and the value of s and the value of s will then be n alpha minus n beta tell me the value like for example here by 2 and of course ms value will exactly same thing with h cross whatever so this is the strict theorem so I can look at any determinant there are space and spin parts which is alpha which is beta and then as then make a comment if it is been adopted or not if it is been adopted this will be the final s state so then I can say it is a single a double a triple a everything for a determinant so for example I give you a determinant 1s alpha 2s alpha and some 1s prime alpha 1s beta 1s prime alpha or even 2s alpha does not matter is this been adopted why yes answer is yes because this please check n beta 1s is contained here so I wrote something else initially so I wrote 1s alpha 2s beta 1s prime alpha is this been adopted no because so hope you will be able to find out from this spin from the space part and the spins one which is been adopted one which is not spin adopted so if I have in general if I have in general orbitals a determinant of this kind phi 1 alpha phi 2 alpha phi 3 beta phi 4 alpha let us say phi 4 beta where phi 1 phi 2 phi 3 phi 4 are different then it is not spin adopted for spin adaptation in this case it is essential that phi 3 and because there are 2 beta orbitals there are 2 alpha orbital they have to be identical there is no other option so phi 3 and phi 4 must be same as phi 1 and phi 2 of course in this case we know it is a beryllium like system so 1s alpha 1s beta 2s alpha 2s beta so obviously it is spin adopted I have just written it juxtaposedly does not matter so that is a beryllium system so it will be singlet or nothing here okay RHF by construction is been adopted now you know that we have made sure that the alpha spin orbitals are identical as the beta spin orbital so the space part of the alpha spin orbitals is identical to the beta so it is by construction spin adopted and of course S is 0 which means it is a singlet state is it clear alright with this I now present to you two different types of hot reform which are very useful particularly when you go to the open shell systems or sometimes a more general closed shell systems or open shell which you have not discussed please remember our discussion has been based on strictly on closed shell system which is restricted hot reform but there are some closed shell system which does not satisfy this where I cannot make this satisfied so in which case I have to worry about more general systems and also open shell systems I am not going to discuss more about it but before I leave hot refog I must say that there are other kinds of hot refogs so one of them is called the restricted open shell hot refog again many of you might have heard this restricted open shell hot refog so this is called ROHF and the second is unrestricted hot refog which is UHF and of course what we have done so far is restricted hot reform so far what we have done is only RHF but there are two important hot refogs which exist one is restricted but can be applied for open shell ROHF another is completely unrestricted so what is the difference between these two is what I will explain now obviously by very name you can see restricted open shell means it is applied to open shells if it is applied to open shells the first important thing is that N alpha is not equal to N beta that is the first thing that we have learnt because closed shell is N alpha equal to N beta of course open shell can have N alpha number of alpha and beta electron same but then they will have different other different properties so let us say now for a particular case that we are discussing N alpha is greater than N beta so then the restricted open shell hot refog is precisely one which fulfills this condition for spin adaptation that means the number of N beta space orbital is or constitute a subset of N alpha space orbit so this is your ROHF so I will give examples and hence it is of course spin adapted by this theorem now that I have already stated the theorem it is very clear that it is spin adapted so N alpha may be greater than N beta it is greater than N beta but then the N beta the lower ones form a subset of this and of course if you write N alpha greater than equal to N beta then ROHF rather ROHF subsumes ROHF which means ROHF is a special case of ROHF so there is nothing great about what we have done in fact if you write this as N alpha greater than or equal to N beta and write this same statement then the ROHF ROHF is a special example of ROHF because N alpha equal to N beta is ROHF is it clear but in general when you write ROHF we assume that the N alpha and N beta different so I am specifically writing that this N beta constitute a subset of N alpha space orbit so let us assume I have four electrons three of them alpha one of them beta so how do I write this so four electron so I must have one as alpha I can have two as alpha I can have three as alpha but the beta must be one of these three so it lets say one as beta that is ROHF I have four electrons so alphas must have different space orbitals by the Pauli principle of course but the beta space orbital must be one of these three so if I write in a very simple chemist notation so this is 1s this is 2s this is 3s then 1s is w occupied this is singly occupied so that is your ROHF and clearly you can see that is a triplet if I write the diagram you can immediately see the triplet and it shows here because N alpha is 3 N beta is 1 so s equal to 1 and it is a triplet state okay and that is the diagram that it shows so it means that whatever is the lower number must be contained in the space orbitals higher numbers then it is called ROHF and I can use now this for open cell system because I can do triplet for example I can do doublet so three electron that is a doublet right it is ROHF yes yes no problem it will be an excited state yeah so it will have it will have a structure like this if it is 2s beta it is still a triplet but it is not an excited it is not a ground among the triplet this is not a ground state that is a different matter we are not talking about which is the lowest energy yes why we need spin adapted because Hamiltonian it commutes to the square very simple so our Schrodinger equation solution must also be eigenfunctional square very often sometimes we cannot enforce but that is because of the approximation we are not exactly solving but in doing approximations if we can enforce it it is obviously more desirable okay just like we like to enforce all symmetry in a problem so spin is another symmetry note that the actual Hamiltonian physical Hamiltonian commutes to square so all exact wave functions are spin adapted is just that because we are doing approximations we are not taking making attention on these so that is the reason we have to bring back and we have to remind ourselves okay on the other hand unrestricted Hartree form is of course by definition has no such restriction so your n beta space orbital need not be a subset of n alpha one of them can be contained others may not so it can be anything so here if it is let us say 1s prime beta which means your alpha and beta are not exactly sitting here but beta is sitting somewhere here call it 1s prime which is slightly different in energy and obviously this is also not there this is a 4 electron system so in this case it is not spin adapted it is not spin adapted and you can actually see if I write this determinant so if I write this 4 you can immediately tell that it is not spin adapted right because you will not call it triplet but if your triplet is just count out number of alpha minus number of beta then it is a triplet but that is not right you count them out when the lower numbers are already contained which means there is a maximum double occupancy so what does ROHF mean ROHF mean do as much double occupancy as possible whatever is open shell whatever you cannot doubly occupy leave them all in the same spin so for example again I come back to ROHF so let us say I have 3 I said 4 1 if I decide that there are 3 alpha spins and 1 beta spins then what is the maximum double occupancy I can do 1 because I have only 1 beta spin so 1 space orbital I can doubly occupy leave the rest 2 with the same spin whatever is not doubly occupied has to have the same spin so I can redefine ROHF in a different way whatever is not doubly occupied must have same spin that is another way of looking at ROHF I decide to write it in this manner because this is strictly more right but the consequence of this that if there are no doubly occupied orbitals then the electron sitting in all those orbitals must have the same spin then it is called ROHF so in a way ROHF ROHF is maximum double occupancy maximum and quite clearly in this case ROHF is a special case of ROHF because I am doing real maximum double occupancy I have 2 alpha 2 beta okay. UHF I do not put any such attention so 2 alpha can be in 2 special orbitals 2 beta can be in another special orbitals and hence UHF is not spin adapted so that is important to realize that the UHF the unrestricted heart rate for determinant is not spin adapted and this is a special problem of UHF at the same time we must recognize that in terms of energy the same problem if I do with UHF and ROHF or ROHF whatever I am again writing ROHF and ROHF in the same box because it is a special case then this energy will be lower than this energy why because in ROHF and ROHF I am making a condition that the beta space part is same as the alpha space part subset in UHF I have no such condition so what does the variational theorem say if you have more flexibility energy will be lower so that is interesting if you want a lower energy and that is my objective then you should better do UHF right but the problem with UHF is that you don't get the spin so you have to worry about what do you want to do okay each of them is approximation of course there is a way to get spin adaptation after doing UHF and that is called projected UHF I will again not discuss all this but there is a method which is called PUHF which means I first do UHF and from the UHF wave function I project singlet, triplet, doublet so by projection operators so this is a highly mathematical in group theory also we do the same thing so this is called PUHF which also exists in the literature so I just thought you should know the name so PUHF in a way is been adopted without projected so this is been adopted so PUHF is against been adopted but moment you do PUHF you lose that energy variational thing because you can't have both because I am now not taking the full wave function I am taking a part of it so I have lost the original part of the wave function okay so I think this these are some of the Hartree fog that you should know again no more details ROHF, UHF equations are actually very easy to derive because we have the general equations the during the spin integration you should have to just take care with alpha and beta not same so you can easily get an alpha so but there will be two equations one for the alpha spin orbital one for the beta and do that ROHF is somewhat more difficult to understand actually UHF is actually much easier ROHF is more difficult and ROHF is understood in many many ways if you analyze the ROHF determinant there is actually two parts to it one is a close part another is open part if you notice all ROHF one is a close part one is open part because close part is the number of beta which is doubly occupied and the rest is open part so many times ROHF determinant is actually said to have a close part and an open part so if I have a determinant form which is close so there is a doubly occupied all of them are doubly occupied and then you have an open part which have open part essentially means all have same spin either alpha or beta all have same spin so this is one characteristic of all ROHF determinant how many how much is the close part how much is the open part that will determine your spin and that depends on also number of electron because together it must be total number of electrons so there is a very complicated Hartree-Fock calculation for the close part open part they are also coupled the Fock operator for the close part Fock operator for the open part they are coupled so ROHF solutions are much more complex than UHF UHF is actually easy however I lose this spin adaptation


Good morning, Hank, it's Tuesday. This video is about looking away, and you may need to look away at some point, and I totally understand that, but I hope that you'll stick with me. So, when Sarah and I went to Sierra Leone, we visited the only hospital in Kono District, Koydu Government Hospital, which has come a very long way. Like, when Partners in Health began supporting the hospital in 2014, there was no consistent electricity or running water. The maternity ward had a dirt floor. There was no emergency department, no blood bank, no functioning x-ray machines. After the devastations of colonialism, Sierra Leone had never had much money to build up a healthcare system, but the system was further weakened first by the Sierra Leone Civil War, and then by a horrifying Ebola outbreak. Five years later, when I visited, KGH was a vastly different place. Clean maternity wards, a functioning emergency department, a laboratory to diagnose illness and plentiful stocks of medication to treat it along with many well-trained healthcare workers. And this is reflected in child and maternal health, more safe deliveries, fewer stillborn babies, and a dramatic reduction in the so-called decision-to-incision time, the amount of time between when the need for an emergency C-section is identified, and when that procedure actually happens. Because these are emergency surgeries meant to save the mother and baby's lives, the sooner the procedure can happen, the higher the likelihood of a positive outcome. And just a couple years ago at KGH, the decision-to-incision time was often over three hours, today it is an average of one hour, eight minutes. But the hospital is still woefully inadequate. There are only 48 maternal beds to serve a population of 500,000 people, and there's no NICU, no incubators or ventilators, there's only very basic equipment to try to save sick kids. And when we were at KGH, I saw a mother with a young child in the pediatric ward, and without going into details, I think the child was probably dying, and I looked away. I looked away because I didn't want to gawk, and because I didn't know what I could do in that moment to help, and because it was awful to see a mother in that kind of pain and to see a child suffering. But there was also something else. I've worked in a children's hospital. I've held the hands of kids as they died.  These deaths are preventable, and they have been for decades. Six years ago, I made a video from Ethiopia called Looking Away about the importance of listening to and supporting people living in absolute poverty. Six years ago. In those six years, I had done some things and given some money, but I had not made a serious commitment to listen better or to focus on the needs of the poor in an ongoing way. So I wasn't just looking at a child who was dying. I was looking at a child who very possibly wouldn't be dying if I had understood earlier the scope of the problem and devoted more of my resources to supporting the people who are solving the problem. And I think ultimately that is also part of why I looked away again. I had understood in the abstract that the maternal and child mortality crises in Sierra Leone were emergencies, but I hadn't been treating them like emergencies, and now I was faced with the consequences of that. The truth was, I wasn't doing nearly enough with the money and power and platform I'd left into to support truly vulnerable people. And I'm still not doing enough, but I am trying to do more, not just with my money and with my time, but also with my attention. On that front, Partners in Health Sierra Leone's annual report just came out, and even if you can't afford to donate to their efforts to build and support stronger healthcare systems for the poor, I really recommend reading it because it is in many ways a very hopeful document. Like one primary health center supported by PIH, the Well Body Clinic saw zero maternal deaths in 2019 for the third year in a row, this in a country where one in 17 women will die in pregnancy or childbirth. So the solutions are happening just not yet across the whole region. Access to mental health care in Kono has dramatically increased. People with cancer or type 1 diabetes, which until recently were considered death sentences in Kono, are now being treated. PIH was also able to hire more staff now, employing over 450 people, 96% of whom are Sierra Leonean. Locally sourced nutritional supplements are decreasing malnutrition, and the maternal and neonatal care team at KGH is hiring 9 new nurses and 4 new midwives. But as PIH's John Lasher told me, Koydu Government Hospital is the only place within many hours, whether walking, driving, or motor biking, where women can receive an emergency C-section, and that puts tremendous pressure on a maternity ward that is just totally inadequate. Which is why PIH and the Sierra Leonean Ministry of Health are working together to build a maternal center of excellence and enicue at the hospital, which will more than triple the number of maternal beds, and also dramatically improve child and maternal care. And it cannot wait, that's why while the Ministry of Health and PIH are going full speed on the planning, we are working very hard to raise the money. In total commitments, we are now close to $13 million, which is amazing, but we still have a ways to go, so if you can make a monthly or one-time donation at pih.org slash Hank and John, we would really appreciate it. Also, every dollar you donate will be matched by generous people from within Nerdfighteria. And if you want to join the matching fund by making a donation of $2,000 or more, there is a link below to that as well. Believe me when I say that I am talking to every rich person who will take my phone calls and doing whatever I can think of to drum up money, but ultimately, I think if this is going to work, it's going to work because thousands of people come together to show solidarity with people in Sierra Leone. I hope from any of you that solidarity can come in the form of donations, but it can also come in the form of listening and paying attention and reading annual reports or telling your friends and family about these efforts. Regardless, thank you for being here. I know that the kind of complex systemic long-term problems that demand complex systemic long-term solutions can be easy to look away from, so thank you for not looking away. Hank, I'll see you on Friday.


Welcome back to my channel guys. It's your girl Coco styles and on this channel We talk about beauty fashion and lifestyle, but in today's video I'm going to show you guys how to shop your closet before actually going spring shopping for dresses Most of us do have a lot of stuff in our closet that we could actually wear either to brunch to church or just a Regular day at the mall. So I Have a few pieces here that I'm going to show you guys The reason why I'm showing you these because they're kind of old, but there are really good length for tall women they're not too short and They're all very affordable. I believe all of these No, three of these are from a sauce Let's see. Hold on This might be from intermix Yeah, so three of these are from Intermix and the other two are from a sauce. So sometimes intermix can be a little bit pricey, but They always have sales. So my recommendation. This is the tip that I always use the moment. I see them have a sale I put everything in my bucket and I give it to like the end of the week of that week Maybe that weekend and they usually go down a little bit more in price. So These are also dresses that I personally think the colors will look really nice on dark skin women I have on a black dress now. I'm not really a fan of black on me. I just never liked black I think it doesn't it doesn't bring me out my skin tone out. So I have one simple dress here and This is by the brand Flounce London and I got this from intermix. Yeah Yeah And where's the price the price is not on here I Actually, no, I did not pay full price for this But It's a simple one-shoulder dress. I'm actually gonna try this on for you guys to see this Maybe I'll do like a split screen that'll be even better So you guys can see it, but it hangs off of one shoulder. Sorry It hangs up in one shoulder Like this very simple if you wanted to dress it down you could but it is like a satin satin material You can throw on the belt with this, but it does come with a simple Tie most people don't like the tie so you can take it off and put whatever belt you want to put on with it This I got this in a size 12 I'm not typically a 12, but I didn't want to chance it. So that's one dress that I got then I got This other one is very it's not my type of dress, but this is more of a Super feminine dress because it has like ruffles. It has like flowers on it and stuff You can win this with a big brim hat With some pumps and you can go to like a little what do you call those big hat luncheons and stuff with your friends It does have some ruching here at the center I believe that you can still wear a belt over this if you wanted to edge it up a little bit and this is by Eliza Jane Now Eliza J does have really nice dresses And this one I got in a size In a size 12 Also Again, I am not I don't I'm not a 12. However, I Always have trouble with getting stuff on my arms because it may be too tight because of my bust area So I always go up a little bit just to make sure it fits me if it is too snug I mean, it's too big. I usually get it take it in so that's the second one from intermix I don't even know where the prices are. I thought the prices were on here And I got another mustard kind of dress and this has like a high low feature very Dell very feminine The back has like a string across cut very low and this is by This is oh actually, this is BCBG. Yeah, but I did get this from intermix and this dress is It's like this weird long tag on the back. I mean this tag is for you to take it off. You can't return it I thought it was inside out and this was actually 250 But I'm telling you I didn't pay 250. I have to have paid half price for this and I got this in a size. I Got this in a large I'm trying to figure out why I got it in a large because oh because I have a BCBG dress And that just felt like it wasn't gonna fit me properly But um, I'll try this on and show you guys how that fits and again These are just colors and some lengths of dresses that I think will look really nice on total women Again, it's hard for us to find things that are not like super short and that look super weird on us So and then I have two other dresses that I got from ASOS something like this I would wear to church Okay, it's not it has like some detailing in it. Um, the back is open However, you can wear a little drape over you and church. It's not like Overly sexy or anything like that because the front is so covered up But this is also another color that I think will look really nice on us more dark skin women And it's a perfect link. I would wear this to church and I would also wear This one to church as well This has like a little belt on the back and this is more of a regular blush This is a different type of pink and it's like a salmon pink and this is more of a blush It's very lightweight. It has like some detailing on it, but it's very very lightweight So I'll show you guys how this fits so I would wear this to church also and then I would wear this to brunch I would if I was going to like a nice little brunch party not like a day party Like a nice brunch party. There's a difference So I'll get these on and let you guys see these Well, like I said, I'll just pop in a video of me next to it while I'm talking and Maybe I'll do a voice over. I'm not a hundred percent sure yet So if I try them on you'll see it. If not, you'll see it pop up. All right guys. I'll be right back Welcome back guys. So I have on the first dress, which is the dress from Intermix. This is a simple satin dress. It has a split give it a little bit of sexy not too much This is something that I wouldn't really do much to actually a lot of these dresses Keep it simple. Keep it very Feminine I think if I start adding belts and stuff to something like this, it'll take away from the feminine part of the dress And just keep it cute and simple. It comes off the shoulder Quite a bit. So make sure you do have on a bra You can wear one strap on one side and then you can just take the strap off the other side Like I said, it's a this is a size 12. So I'm just saying I'm not a big girl by all means like No way down here But it fits me a little Fitted so I'm glad I did get a 12 in this one So but if you are hippie you may have some issues with something like this But I typically wear dresses that are this length because I don't know I just feel comfortable in them and I think they look better on me anything shorter. I probably totally will feel uncomfortable Fidgeting and stuff. So something like this I would wear. So this is just the first dress I'll turn around let you guys see the back Comes down like that. It's it's it's pretty fitted but not to the point where I can't move in it it does have a little stretch and I have on nude shoes right now, which are some nude sandals from Stuart Weissman now I have big feet. I wear size 11 and I have wide feet. They do carry wide and They're barely nude. I Believe those are called Bailey news and I forget what these are called. But these have like a block hill. They're comfortable But they are a little pricey in my opinion what I didn't like about the black ones were that the first time I wore them the bottom was like Pretty much like I wore them for like two months straight. So If you did buy something like this just think about putting on a sole protector so you don't have to do it layer on but This is the first dress and I think it's simple. You can wear black sandals if you want it I have on nude for now because that's what I have over here and If you want it, you can just do a simple clutch I probably wouldn't do this but it's very bright So if I wore this I would wear black shoes because it does have some black detailing or I can probably wear My little LV clutch And just keep it simple. I'm like I said, you can wear black with this if you want it I'm a lot of girls wear like the clear shoes. You can also do that if you want it So it can look seamless, but I have them on with this right now Well, I have it on with this right now So I'm gonna just try on the second dress and let you guys see it and I'll be right back Sorry guys, I had an issue with my last clip. It is completely lost. So I apologize for This clip showing in portrait mode, but here is the dress All right guys, so here's the what is this the third dress? Okay, now hear me out I went through a phase Where I wanted flowers and stuff, I have no idea why but I actually really like the color in to my surprise I believe this is a 12 also and it's fitting me just perfect. So So that means to me anything from intermix Actually, I don't even want to say anything from intermix because this was a BCBG dress and this is Eliza J So if you got Eliza J and you are between 190 195 and 200 and a slim a 12 will fit you perfect So this is very Girlie to me. It's not really my cup of tea However, I just wanted to still show you guys because I know flowers are in this this spring So if you want it you can wear something like this It does have some ruching here So you can put a belt if you want it if you can wear white shoes with this I have a new I kept the same new shoes on you could put a white belt on if you wanted like a big belt To take away from this it has ruffles and stuff guys and the fact it's just It's just the opening in the back here nothing too nothing too over the top but um, I Probably won't wear this. I may sell it. So if you're interested, let me know But I'd like I said I do like the color blush Midgrain Ivory creams Like nice news. They look really nice on deep or darker skin even Mustard looks really good on us Red looks really nice on us. I'm really trying to stay away from all white It's okay, but I like the ivory just a tad bit more And for me, I like gold jewelry. So but anyway I'm gonna get the last two dresses on the last two other ones that I said that I would wear to church so if you are a person that goes to church and you don't really know if you can wear like certain colors or I think these are a perfect length and they're still very stylish. So I'll get these on in a second and I'll be right back Welcome back. So All right guys, so here's the fourth. Oh this dress Because I had on the black one. All right, so this is something that I would personally wear to church I think it's the perfect length for church now We all know sometimes they look at the young people because they have one short skirts and stuff But I think this is actually really cute and enough to transition over to like a brunch with your girls after if you wanted to I'm gonna have like a little bit of detail in here and the back like I said may not be for church, but you could always just throw on a Shawl or something while you're in church, which is usually cold anyway. So and this is like a Salmon color, maybe it's not really blush, but I also think this is a very pretty color also So let me know what you guys think about this one. Would you wear something like this to church or to a dinner where You know you have to kind of be Not formal, but you know not too revealing I guess But I think this is simple and it's cute I'm very playing when it comes to some things sometimes I do believe less is more I don't think you always have to jab up an outfit to look Presentable nice and pretty and classy. So that's my tip. So this is it for this dress I'm gonna get on the last one and then we'll be out of here. We'll be done for Today's video. All right guys. I'll be right back All right, so this is the last dress guys are so all right guys So this is the last dress. This is the other one from a sauce. I'm going to tell you guys that this dress I Got in a size 12 a sauce brand itself Usually run rather large this wasn't like that So I would recommend when you're buying dresses from them and this is actually from the tall set. Oh, yeah This is from the tall section It sure is that's probably why it fits the way it fits at the um At the back. Sorry. This is a US 8 US 8. Let me just triple check the black one Yeah The black one that I had on also. I'm sorry guys. I told you guys at 12, but that's a UK 12. It's a US 8 The rest of these are from intermix and oh Again, this is also a US 8 a UK 12 they didn't made in the UK And then the BC BG one That's a large and then I believe the Eliza J is a large also as the 12 actual 12 Okay, so Here's the last dress. This is something that again, I would wear to church and This kind of reminds me of like Easter colors like back in the day when I was like 12 Which was a very very long time ago But um something like this like I said you sometimes you do not have to always add on to something To jazz up the outfit sometimes the dress is just enough. I think this is simple I wouldn't want to do too much to it because I think it's a very pretty color and I think it's simple It has like it almost looks like corduroy, but it doesn't feel like corduroy the back has This time which I tied it really bad, but you can tie it into like a bow or something or let it hang Like with a knot or whatever if you don't want it to like You know be bulky in the back It does zipper all the way up But I think this is simple, but um, that's it guys So if you like videos like this kind of like shop your closet before you go shopping Videos let me know I can do more of this. Um, I am gonna have some maxi dresses coming soon I'm just trying to make sure that I get nice ones because it is starting to get really warm where I'm at So if you like videos like this and you stay to the end of this video Please comment below and please comment Coco. We see you and don't forget to hit that like comment and subscribe And I will see you guys in the next one. Bye


Hello, I'm the citizen. And I'm the government. Does this feel like kind of a dated parody to you? What? No, this format is the new hotness. No, Apple hasn't been doing Mac PC ads in over a decade. You know I still have laws on the books from the 1800s making it illegal to offend a horse, right? Right. Anyway, if you've got a better idea, just fax it to me. Cool, yeah, I will. Hello, I'm the citizen. And I'm the government. Did you see my mandate? You didn't win by that big of a margin. I can't wait to ran through my legislative agenda with my enormous mandate. Okay, calm down now. A third of eligible voters voted for you.  You want to see a picture of my mandate? No. Sent. Gross. Hello, I'm the citizen. And I'm the government. It's peacekeeping.


Welcome to Psychic Medium, Tony Green. I am Tony Green, the Psychic Medium and Channel. So, so many things today. Oh my goodness. I first thank, thank you everybody for being here with me, for being part of the show, for calling in, for joining in on YouTube. If you want to join the show live, I'm here every Monday and Wednesday at noon for a minimum of 30 minutes, most of the time 60 minutes. I take callers, if you want to be one of the callers, the number is<PHONE_NUMBER>. Also, if you want to join on YouTube where we have chats going on, go ahead, go over to YouTube, put my name in, and join in the show that way. However you join, it's perfect. So, this show does air live on TV, WSES, RUQ, and a number of other streaming channels along with many, many, many, if not almost, I think most podcasts. The show is now on almost like a lot of podcasts. Rude Rangers TV distributes the show on all of their venues, WSES distributes the show on all of their venues. Okay, having said that, thank you, Rudy, and thank you, Scott, for all of the work that you do for me and the show. I just adore you guys so much. Next, a couple little things. First, I said a while back, quite a while back, it was when a jackpot was pretty high, and I said, we're going to see a billion dollar jackpot. And then after that, the lottery is going to come out with a new game called, that starts like basically at or near a billion dollars. Well, today, I woke up and I looked at mega millions and it's at a billion. I just, I wanted to see if I won last night's jackpot, apparently I did. But it's at a billion dollars. And no, just because I'm a psychic, it doesn't mean I should be able to win the lottery. I wish it were that easy. I wish, right? But it's at a billion dollars. Now, I'm not sure if this is the first billion dollar jackpot, but there we go, folks. We are at a billion dollars. Now, I say, give it about a minute, and there is going to be a new game from the lottery, and it's going to start at a billion. Having said that, the next thing that I want to go to, if you would like to connect with me, if you would like to connect with me via social media, there's a link in most of the descriptions that will get you to all of my social media pages. However, I am on TikTok, Instagram, Facebook, and of course, YouTube. TikTok, mostly I just do healings for people. The healing this morning, it was for depression. So if you or somebody you know is suffering from depression, please go over there and direct that clearing to whomever needs it or use it for yourself. The other page I have on TikTok, I love, it's called Live Beauty Reviews, and it is, it's live.beauty.reviews.  Yes, let the fun begin. And then next, what else? What else? I have little notes here. Okay, I'm going to just say, I'm just going to, I'm going to say a little something about relationships here. First and foremost, when I talk about relationships on the show, I understand how difficult a relationship dynamic is. When we are in a relationship and we are trying to get through our own personal life, and then we have somebody else's life has entered and trying to mesh those two lives and then understanding each other and working with each other and making it successful is not easy. And society or whatever has led us to believe through romantic comedies or fairy tales or whatever, that relationships are very different than what they are. I've seen people with really amazing, seemingly healthy relationships, but they work at it. They have agreements with each other and they really, really work at it. And then I see really, really chaotic dysfunctional toxic relationships. One of the problems that I see with relationships today is we're so busy trying to figure out what's going on with the other person that we're not making logical, healthy choices for ourselves. Let me repeat that. We're so busy trying to figure out what's going on with the other person. Are they doing this? Are they doing that? Why are they doing this? Why are they doing that? We're in a state of confusion. Are they cheating? Are they not cheating? If there's not another person I don't want to break up, I don't have evidence, I need evidence. Are they lying? Are they not lying? There's a state of confusion with one of the people and because of this state of confusion, you are stuck. You become obsessive in your thoughts about this person and you can't move forward. You don't want to break up in case you're wrong. You don't want to have it continue the way it is. And here's what I'm going to say. It really doesn't matter if they're cheating. It really doesn't matter if they're lying. If you are in a state of confusion, illusion, delusion, if you are in a state that is putting you in a place where you have to wonder if your relationship is secure or stable, get out of that relationship at least temporarily. Sometimes you can't fix what's broken from inside of it. Sometimes you have to get out and then the other person's willing to work on it or talk or have different conversations about it. But the bottom line is here's the bottom line. Most of us are normal, healthy, logical people regardless of what our exes say about us. Okay? If we do not feel stable or secure, we do not belong in that situation. We do not. And let me say this. You don't deserve that. You do not under any circumstances deserve to be in a situation where you do not feel safe. And that doesn't mean only physically.  Who didn't have a chaotic whatever upbringing? Who didn't? Okay?  If you do not feel stable, like they're in this for the long haul, you have every right to excuse yourself from the situation. And furthermore, you do not deserve to be in a situation that has you feeling anything other than loved, safe, and secure. That's the minimum a partner should bring to the table if they're wanting a long term relationship. Now here's the problem. Many of us are in relationships hoping the other person is going to get there. We hope something's going to happen. We hope they're going to finally see we're the one. They know we're the one. They wouldn't be dipping their diddly in our do if they didn't think we were all that. Trust me on that. They wouldn't. Okay? We're just hoping they're going to get to that next step and want and a person wanting a full on relationship might not have anything to do with you personally. It's just where they are in their life at that time. So don't take it personal and never ever try to win someone.  Now, having said all of that, I'm about to start taking callers. If you'd like to call in and be part of the show, the number is<PHONE_NUMBER>. Nope. That is not the number. Do not call that number. That is my business number. The number is 845. Wait, what's the number?<PHONE_NUMBER>. The other number, 414, that's my business number if you want to schedule an appointment, a private appointment. If you want to schedule a private appointment, you can also go to my website, find out all the information you need. My website is t-o-n-i-g dot i-m-f-o. I'm going to go to start taking callers. I want to say hi to everybody joining me on YouTube. Lolly, hey, Chico, what's up? And Mecca, I think her name is... Oh, no, Nika. I'm sorry, I'm massacring your name. I'm so sorry. Patrick, Caroline, Heather, I want to make sure I'm not missing anyone. Thank you so much. And Cindy, I know you listen to my shows. And Jess, I love you guys so much. And Jana, there are so many people. Matthew, thank you guys so much for being a part of the journey. I'm going to go to the very first caller who is 586. 586, how can I... What's your name and where are you calling from? Ken, how are you? Good, sweetie. Thank you so much for calling in today. So what's your question and what's your question or connection that you would like to work on today? I would like a connection from the other side. And the name of the person you would like to connect with? Lele. Lele. Okay, so the energy hits me and it's like, she's like, where's the party? I don't know if this is her energy or somebody else's energy, but the first thing that hits me is a lot of fun. And where's the party? Is this the right energy that I'm connecting to? Okay, thank you for that confirmation. I just want to make sure I have the right... It's very normal that more than one person will step in sometimes, but I want to make sure I have the person that you want.   Okay, I think it's that song. But I think if you know it, you know it. If you don't, my singing is not going to get you there. That's for sure. Okay, so the first thing she is saying is it was a really wild ride, but she doesn't have any regrets. She wouldn't change a thing except for maybe that one thing. And then she just busts out laughing and she is saying that she is so... She is so... Okay, now she's getting serious. Now she's getting a little serious. And then she goes, oh, she goes, girl, if I knew, if I knew then, oh. And the way she's saying this, she's not saying what that is, that if she knew, but I know you know, because if she needed to elaborate, she would. She is saying that she... She's really grateful for everything. Looking back, looking back on everything as it happened and how it happened, she's saying she finished her job. Now, that doesn't mean a job here, one specific job here on earth. That means she did what she came to this plane to do. So she finished her job. When that job was, she finished it. She did it job complete. She goes, I can check that off my box now. And she wants you to know that she is... She's sorry. And as she's talking, you know, folks, I need to let everybody know, as loved ones are talking and they're saying things, I'm also asking them questions. So some of the information I won't get because it's just they know you, the person it's meant for knows what they're sorry for or what that one thing was, like you're going to know that where they don't need to announce it to the whole world. She's sorry. And she... Whatever that means for you, for you personally. Yeah, I'm not sure what the message is. She says she should have, she said, okay, she's saying she should have turned around, she should have turned around, she should have turned around. Now, I don't know, and I have goosebumps on my legs on that one. I don't know if that is a literal, she was on her way someplace and she should have turned around. Or if that was she should have turned her life around at some point, or she should have turned around her, for example, finances, she should have turned around her relationship, she should have turned, but she should have turned, she should have turned some... If you go in many ways, yeah. Exactly. And for her, for her life, for each person's life, that statement's going to be different, but for her life, that statement has a specific meaning and if you think about it, you're going to know what that meaning is. And then it might have been like a relationship thing because she keeps singing, I keep hearing that song by Bonnie Tyler, turn around, every now and then I get a little bit lonely and I never see you coming around, turn around. Oh, Jesus, not today. Oh, that is like way back when song. So she should have turned around, she should have turned around whatever that means or whatever that meant, she should have turned around. Now she wants to go into confirmation, onto some confirmations, like when you see that means she's around and that she's letting you know she's there. So one of her confirmations are these yellow lilies or not tulips, like they're the pointy, like they're like lilies, I think they're lilies, I can see them, lilies. I'm just going to go with the yellow lilies. And even if they're white and she's saying, now the next thing I'm, oh, she's throwing flowers at me so fast right now. The next thing I'm going to say is, or, or Leander, or or Leander, or somebody tell me how to say this word, O Leander, I don't know, I don't, I don't know this word. I don't know what this is. I think, I think it's or Leander, or Leander, or I think it's, I think it's like a poisonous flower. I don't know. I don't know. I just, I don't know. I don't even know if I'm saying it right. And then the next thing she, okay, she's flowers, any flowers, but especially those, those crazy flowers. And then the other thing that is a confirmation for, for, from, oh, I love that one. And I know why. Okay. So bunnies, whenever you see a bunny, any kind of bunny, she is letting you know she's over. Well, she's letting you know. I have a bunny. Oh my gosh, really? Then that is it. Yes, I have. I love that. So whenever you see, I have my bunnies. Okay, I'm sorry. Go ahead.   No, I said I have my bunny for about three years now. Okay. So that's so funny.  So please know that. Yes. The word. So one of my YouTube, yeah, I'm sorry. Go ahead. Love. Oh, I'm sorry. I hope she's be there during my pregnancy when I deliver my baby. I hope she's be there around me during my pregnancy. Yes, I'm hearing. Yes. And I'm hearing. Okay. So a lot of people, one thing I always say to people is when a baby comes in, they've been on the, obviously the other side and they, they're getting acclimated and their loved ones on the other side help acclimate them and teach them, you know, about us in the life they're coming into.  This, this child, like if I were going to choose a word, but it wouldn't be the right word. It would be a little bit snarky, but I'm going to just choose the word funny.   So please know that that that's going to be coming in. And yes, she will be there. She's saying with a catcher's mitt. So yes, okay, those are her messages for you. That's what she wants to say. That's what she wants you to know. And she wants you to know that she's, she's going to be hanging around as, hanging around like as, as much as she can, but even more than before. Like, so she's just always going to be home. So I really want a connection with her. I really want to know she is. Yeah. And she's also letting me know she's going to help you like through the first six months, especially the first three months, she's going to be helping with the baby. Okay. Thank you. Yes. And the, the name you are so welcome. Thank you. And the name of the flower Patrick from, from YouTube said Oleander. Pretty but poisonous. It is poisonous. Oleander, very pretty, but poisonous. And she goes, like me, pretty. So that is, thank you so much. Thank you. Be a little girl. That would be nice. Yes. Yeah. You know, I, I don't like to give out the sex of the baby, but I think you're going to be pleasant, Lisa. That's all I'm saying. Just saying. Okay. Okay. Okay. You have, thank you again so much for calling in. No, thank you. I really appreciate you a lot. Oh, thank you. Very helpful. And I will go back and listen to the video on YouTube. Thank you. Oh, thank you so much. Do you have a good day? Okay. You too. Thank you. You're beautiful as well. Oh, thank you so much. Thank you. Thank you. Thank you. I'm going to go to the next caller in just one second. So please hold on. I want to say hi to some of the new people on YouTube. So it's any CKA, sorry if I mispronounced that. Please no, I did not mean to. And she called in a few weeks and haven't been able to speak to you live. I hope you can answer my question. I wanted to know if there are any messages for spirit mitt for me. Okay. Can you please narrow that down a little bit if there's one specific person that you want to connect with on the other side and then just print, just just print. And I'm sorry, I can't pronounce names very well, guys. Okay, I'm going to go to the next caller. And you guys, it's always easier to connect with me through calling in and asking the question. And the number is<PHONE_NUMBER>. The people on the call log have been waiting for 15 minutes before the show even started. And I always feel like I really want to give my attention to them for that reason. I'm going to go straight to 312. 312. And it's easier, honestly, for me to connect to energy when I'm talking rather than everybody in a chat. So 312, what's your, what's your name? Where are you calling from? Hey, Brian, how can I help you today? Hey, I was laid off from a job that I had been working for a year. We applied for a dream job of mine and I waited here back. And I really want to know when do you see it great to come up for me? Okay, the first thing I hear is September. Now a breakthrough doesn't necessarily mean that that's when the job's coming in. That might be when you actually are starting the position. I feel like September is going to be a really good and powerful breakthrough month for you all the way around. I feel like during the month of August, there's going to be a lot of communication or what they're calling chatter back and forth. I feel like you are going to have some opportunities coming in all through August and then in September, something you're going to be, you should be, you could be, depending on how everything plays out, you could be starting something new. I do feel like, hmm, okay, I feel like during the month of September, you're going to get two, two, possibly three offers. Now, I always, always say this, even if it's not what you want, just take it. It takes the pressure off of not having it and it puts you in a new energy. And, but one of the offers is going to really stand out to you and be what you want. I feel like that offer starts sometime in August, like the interview process starts sometime in August, and then it, by September or in September, you should be with that, that particular offer. But there will be maybe two other offers. Either way, September is your breakthrough month where everything starts moving forward again.  I get a yes, we're going to clear that. All that's causing us to be stuck has been cleared, released and healed. I get a no, let's clear that. Thank you so much everybody for joining me today. I will be back Monday at noon to give you another hour of this. Thank you so much. You guys have an absolutely amazing weekend.


Welcome to ToveyTV to Everton News Daily. Everton have released a statement to announce changes to the board of directors They announced that Chief Executive Officer Denise Barrett-Baksen-Dale Chief Finance and Strategy Officer Grantingles and non-exec Graeme Sharp have left their roles at the club. The outgoing directors have worked tirelessly over the recent months to assist with the preparation for a transition to a new board. The club is very appreciative of this generous accommodation, which is both characteristic of them and entirely in the spirit of the best values of our club. A statement from the department director reads, we have been fully committed during our time here. We are disappointed to have made the decision to leave Everton. We have worked tirelessly alongside the chairman in what has been a challenging period to deliver some of the most significant projects in Everton's history. Projects that will safeguard and sustain the commercial future of the club for generations to come They go on to say we are proud of the commercial growth project progress during our time on the board including the construction of the club's new waterfront stadium, the construction project the largest currently on site in the UK is progressing at pace. Yeah, Bill Kenwright has come out and said also that this has been a great board who have all worked tirelessly for the club no matter what the circumstances. My relationship with Denise has been one of the as known one of the closest in football. I thank her for many achievements particularly her magnificent work in respect for our new stadium. I think it was pretty inevitable that would be relieving. I think it got to a point where they couldn't really carry on no matter what they did there was always going to have that you know the history of where Everton have finished under this board hanging over the head so and it seems like even though they've been there for no for years now they don't seem to be learning from the mistakes and they've been given you know plenty of opportunities and we seem to get worse and worse every year you know finishing 16th and then you know finishing 70 the next year you know it's not good enough and they keep making the same wrong decisions. You know there's a famous court by John by John Blay and change the people or change the people meaning you know the people can't change in the job then get rid of them get new people in and I think this was always going to happen it's interesting that the chairman hasn't stepped down whether that whether he will I don't know or whether a chairman needs to be in place to you know at all times or something else I don't know how it works but I imagine he'll go as well I'm not too sure at this point but yeah I'm going to leave you with bars for the rest of the news. Carlo Ancelotti is to Sue Everton is being reported today the Italians lawyers made the submission to the High Court on Friday and this relates to general commercial contracts and arrangements the 64 year old left Everton to join Rail Madrid despite signing a four and a half year contract with the with the trophies and is held in high regard by Evertonians lots of blues devastated when he left Everton. Carlo has gone on record saying he's still in Everton and he wants the club to do well so this will come as a little bit of a kick in the plums. There is the latest the latest in a line of press that doesn't really reflect well on Everton doing things as a business he's entitled of course he is of Everton owe him money and having fulfilled parts of a contract then he's well within his right to take the football club to court but we'll have to wait and see the outcome of this one but Carlo Ancelotti is set to sue Everton in the High Court. Everton will receive £2.5 million due to Manchester City winning the UEFA Champions League on Saturday night City beat Inter 1-0 with a 68th minute goal from Rodry to give them their first ever Champions League it helped Pep Guardiola win the Champions League again this time in a different club a first-taste and successor Barcelona is now achieved at Manchester City but the windfall is due to the contract that Everton signed with City which took John Stones from Gooderson Park to the Etihad back in 2016 for £50 million the defender come midfield player as he was at the weekend there's been a fantastic form for City of Leite he was absolutely majestic on Saturday night he's been brilliant in that role in midfield the role I always thought he could play he seemed more of a that type of player in the centre back but he's been brilliant for City and just got better and better for them culminating in that Champions League win at the weekend which means Everton are £2.5 million better off that might go up the door to Carlo Ancelotti unless they short that out in typical Everton style Damari Grey is the subject of interest from Fulham that is according to reports today Grey joined Everton in July 2021 from Bayer Leverkusen he's got 12 goals in two seasons for the Blues he joined for £1.6 million from the German club other clubs are thought to be interested in the Winger who turns 27 at the end of this month the rain reason that people are saying Grey may move on is due to lack of starts at Everton under Sean Deitch Grey was in and out the side and once Sean Deitch came through the door in January no no fears being mentioned as yet Grey still under contract at Everton and Everton have again today been linked with a move for Lyon striker Moussa Dembele the 26-year-old is out of contract at the end of June if Jainas Fabrizio Romano stating today that Everton have the proposal as along with Galatasra and a couple of other clubs know exactly what Dembele is looking for and I've got a good opportunity to take him Everton have been interested in taking him in January as well couldn't quite get that one over the line but the opportunity is there to take Dembele on a free at the end of the month. Dembele has scored 70 times in 172 games for Lyon since joining them from Celtic Everton I thought I want at least two centre forwards in this summer to cope with the injury injuries we had last season which obviously Dominic Ardolun suffered greatly with injuries last season but Sean Deitch wanting two strikers in as well as a wing get midfielder and the left back apparently this summer that is if for the news day to give the video a thumbs up and subscribe if you haven't see you later