What is Gaussian: Definition and 771 Discussions

Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70. It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16. Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.

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  1. A

    A one-dimensional Gaussian Wave Packet

    Homework Statement I have been given the function \varphi_{G}(z,t) = \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ e^{-\frac{4(k-k_0)^2}{\Delta k_0^2}}e^{i(kz-\omega t)} and been told to do the integration and then to specify the phase and group velocity of the wave package. I...
  2. P

    How to evaluate this double integeration of a gaussian function?

    how to integrate this equation? ∫∫e^-x^2 e^-x'^2 / |x-x'| dx^3 dx'^3 lower limit is 0 and upper limit is inf for x and x' the result is an error function. But I would like to know the details of the process of integration. some one suggest me to change the variable to x+x'=u...
  3. V

    Potential due to anisotropic gaussian distrubution

    Hello All If there is spherically symmetric gaussian charge density (http://en.wikipedia.org/wiki/Poisson%27s_equation) \rho(\mathbf{r})=\frac{q_{i}}{(2 \pi)^{3/2} \sigma^3} e^{- \frac{\lvert r \rvert^2}{2 \sigma^2}} then it will have have potential \phi(r) by solving Poisson equation...
  4. K

    Scattering of a gaussian wave packet at a potential

    Homework Statement start with the wave function \Psi(x,0) = Ae^{-cx^{2}}e^{ikx} where A,c, and k are real constants (and c is positive) i) Normalize \Psi(x,0) ii) Determine \Psi(x,t) and |\Psi(x,t)|^{2} Homework Equations The Attempt at a Solution I normalized it to get...
  5. R

    Help with Gaussian integration problem please

    Help with Gaussian integration problem please :) Homework Statement Compute the improper integral \int^{\infty}_{-\infty}x^{2}e^{-x^2}dx given \int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}. Homework Equations Just the rule for doubly-improper integrals I guess...
  6. C

    Gaussian distributed falling drops on a non-orthogonal plane

    Hi all, I'm considering a problem of a bunch of particles (say electrons) arriving on a plate. If the plate is orthogonal to the direction of motion of the particles their distribution (on the plate) is a gaussian distribution centered in a point (x,y) of the plate with standard deviation...
  7. J

    Integration of a gaussian function

    Hey, all. I need help computing a particular integral that's come up in my quantum mechanics homework (Griffith 1.3 for anyone who's interested). It involves integration of a gaussian function: \sqrt{\frac{\lambda}{\pi}}\int_{-\infty}^{+\infty} \! {x}{e^{{-\lambda}{(x-a)^2}}} \, dx Where...
  8. M

    Ideals of the Gaussian integers

    I'm working on an exam that Michael Artin once gave, where one of the questions is basically, Consider the homomorphism from Z[x] to Z[i] given by x --> i. What does this homomorphism tell you about the ideals of Z[i]? So far I haven't come up with anything. I know in advance that the...
  9. M

    Sampling of a gaussian distribution

    I have a 3d data cube. For every point I measure the property A which is a gaussian variable of mean m and variance s and it's also a function of the density d at every point. A(x,y,z)=f(d(x,y,z))e^(-(X-m)^2/(2\sigma^2)) X is a random number. Now let's say I want to sample the...
  10. E

    Matrix notation - Two jointly Gaussian vectors pdf

    Hello I am having trouble deriving using block matrix notation the conditional pdf of two joint Gaussian vectors: I assume that it just involves some re-arranging of eq 1 (attatched) but am unsure if taking the inverse of the resultant matrix in eq 1 is valid and if the order of...
  11. C

    How can I create a Gaussian curve with given data?

    Hi, My understanding in this is limited, but I'm trying to create a set of Gaussian curve response functions that will be convoluted with some signal data. I've dug all around the internet/textbooks but I'm having trouble finding a way to create the curve with the data I have. All I know is the...
  12. QuarkCharmer

    Gaussian Function in Statistics

    Homework Statement Not really a homework problem, but something that I was curious about. I was thinking about how a calculator finds the area under the standard normal distribution, and I started to assume that it most likely has the antiderivative for the function that makes the standard...
  13. E

    Gaussian random variable joint density with discrete pdf

    Hi all, I am having trouble with the concept of joint pdf's. For example - a set Z1,Z2,...ZN are each gaussian rv. Let Z1~N(0,1), let X be +1 or -1 each with probability 0.5. Z2=Z1X1, so Z2 is ~N(0,1). (I assume this to be As Z2 is just Z1 multiplied by a simple factor, an instance...
  14. H

    Four-point Gaussian quadrature rule

    I need to use the four-point Gaussian quadrature rule to do some intense numerical calculations. Could anyone link to this page where it's written out explicitly over an [a,b] interval. I haven't been able to find it, I'm trying to derive it now but it's crucial that I'm 100% correct. I haven't...
  15. F

    Gaussian beam and focusing versus geometrical optics

    Hello Forum, if we take a Gaussian beam whose waist occurs at the front focus of a positive lens, we will see that the Gaussian beam will have another waist at the back focus of the lens... That seems to be in contradiction with what happens in geometrical optics: if we place a point...
  16. V

    Integral of Bessel function, square root and gaussian

    Hi! Does anyone know how to solve the following integral analitically? \int^{1}_{0} dx \ e^{B x^{2}} J_{0}(i A \sqrt{1-x^{2}}), where A and B are real numbers. Thanks!
  17. S

    Approximation of skellam distribution by a Gaussian one

    Hi, everybody Let n_1 ~ Poisson ( \lambda_1 ) and n_2 ~ Poisson (\lambda_2). Now define n=n_1-n_2 . We know n has "Skellam distribution" with mean \lambda_1-\lambda_2 and variance \lambda_1+\lambda_2, which is not easy to deal with. I want to find the Pr(n \geq 0) . Is it...
  18. W

    Interpretation of Gaussian Kernel

    Hello. In a software application I am attempting to smooth a data set by convoluting it with a discrete Gaussian kernel. Based upon information garnered online, I've been using this Mathematica command to generate the kernel: kern = Table[Exp[-k^2/100]/Sqrt[2. Pi], {k, -range, range}]; where...
  19. S

    The PDF of the exponential of a Gaussian random variable

    What is the PDF of the exponential of a Gaussian random variable? i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)? Thank you!
  20. I

    Charged Plate w/ Gaussian Cylinder: Integral E*dA

    1. Consider a uniformly charged conducting plate with an infinite extent, where the sur- face charge density is surf . We introduce a Gaussian surface as a cylindrical pillbox of ra- dius a, half-way submerged into the surface. The figure below shows a chunk of the infinite plate and the...
  21. T

    Calculating Width of Gaussian PDF(x) Given Sigma & N

    Hi Experts, I'm working in industry and have an application requiring some expert knowledge on statistics/probability. I have a probability distribution function (PDF) for a Gaussian random variable. I know the standard deviation of the PDF. I also know total number of experiments conducted...
  22. S

    Stability of the gaussian under addition and scalar multiplication

    If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent , then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way, for the mgfs way is it just, mgf(ax+by) =...
  23. S

    What is the convolution of two independent standard gaussian distributions?

    Hello, my question ; Suppose X1 and X2 are independent random variables, each with the standard gaussian distribution. Compute, using convolutions the density of the distribution X1 + X2 and show that X1+ X2 has the same distribution as X * root2 where X has standard gaussian distribution...
  24. S

    Raw moments of Gaussian Distribution

    I'm wondering if there was a table of moments for a Gaussian Distribution, I found one up to the fourth moment U \sim N(\mu, \sigma^2) E[U^2]=\mu^2+\sigma^2 E[U^3]=\mu^3+3\mu\sigma^2 E[U^4]=\mu^4+6\mu\sigma^2+3\sigma^4 I'm doing a problem right now and i need the 8th moment.
  25. M

    Gaussian curvature for a given metric

    Homework Statement Assume that we have a metric like: ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2} where r,\theta , \varphi are spherical coordinates. f,g and h are some functions of r and theta but not phi. Homework Equations How can I calculate Gaussian curvature in r-theta...
  26. V

    Calculating Spectral Weight of a Gaussian Curve

    Can you tell me what spectral weight is? If I have a spectrum that I measure, how can I get spectral weight? For simplicity, assume that the spectrum is a Gaussian curve. Thank you so much!
  27. D

    Calculating percentile ranks of a Gaussian distribution

    I have been calibrating a sub-micron particle sizer with 1 um (1000 nm) standard. After testing the standard, the test results on my print-out: (X25 = 812.7 nm, X50 = 977.7 nm, X90 = 1389.0 nm) According to USP, the limits are +/- 6% of the reference standard values for X25, X50, and X90...
  28. E

    What Is the PDF of a Circularly Symmetric Complex Gaussian Vector?

    Hi, Suppose that an n-dimensional vector \mathbf{z}=\begin{pmatrix}z_1&z_2&\cdots & z_n\end{pmatrix}^T is characterized as a zero-mean circularly symmetric complex Gaussian random vector. What is the distribution (the probability distribution function PDF) of this vector in both: complex and...
  29. D

    Showing the uniform convergence of a gaussian function-like series

    Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...
  30. T

    What Is the 2D Equivalent of a Normalized Gaussian Function?

    This isn't a homework question at all, but I thought I should try not to pollute the general math forum with it. At work, I am trying to create an artificial distribution of a quantity. It has a peak at a certain location in 2D space (x',y') and then falls off with a gaussian profile in every...
  31. N

    Solving pde with gaussian function

    I just developed this model to describe an ecological process but have trouble solving the equation. My first question is: is the form even analytically solvable? And if so, what steps / references should I resort to? 'a', 'delta' and 'sigma' are all constants. *Current reference book...
  32. M

    What is the Advantage of Using a Gaussian Distribution of Noise in a System?

    what is the advantage of using a gaussian distribution of noise (white/colored) in a system over any other distribution ??
  33. T

    How Does Charge Distribution Affect Gaussian Surface Flux?

    Homework Statement A metal sphere of radius a is surrounded by a metal shell of inner radius b and outer radius R, as shown in the diagram below. The flux through a spherical Gaussian surface located between a and b is 1.20Q/εo and the flux through a spherical Gaussian surface just outside R...
  34. S

    Background for Gaussian Integers?

    Hi, I was wondering if having some training in algebraic number theory is a must for even starting to work with Gaussian Integers, or one can work with them with some knowledge of abstract algebra, like group, ring and field theory knowledge (i.e. 1 year of undergraduate abstract algebra)...
  35. M

    Gaussian Surface, Coaxial Cable

    Homework Statement A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial cylinder with inner radius b and outer radius c. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per...
  36. A

    Make use of Gaussian symmetry?

    The question is: for a gas of weakly interacting molecules show that <v_x>=0 where <v_x> is the average velocity in the x direction. the probability of a molecule having a velocity v is given by: p(v_{X})=\sqrt{\frac{m}{2\pi kT}}e^{-\frac{mv_{x}^{2}}{2kT}} The above is a Gaussian...
  37. F

    Gaussian Beam focusing with lens

    Homework Statement Find the condition needed for a gaussian beam with waist a distance d0 from a thin lens to be imaged (so its geometric image is at di from the lens as per 1/f=1/di + 1/d0) so that its new waist occurs at di, the geometric image position. Homework Equations...
  38. X

    What is exactly pivoting ( in Gaussian elimination)?

    In terms of programming. Like, let's say you have F1+F2+F3+15=0 and you move the 15 to the other side, just as you're supposed to, which would be F1+F2+F3= -15. Is that what they mean by pivoting? The code moves the 15? thanks
  39. A

    Product of gaussian random variable with itself

    Hi, I am interested in the product of a Gaussian random variable with itself. If X is Gaussian then what is X^2? We know that the resultant variable of the product of two independent Guassian variables is still Gaussian but I am afraid that this is not true when you multiply it with itself. Is...
  40. I

    Electric Field of Infinite Plate: Why Use Cylindrical Gaussian Surface?

    I'm trying to show that the electric field for an infinitely large plate is E = \frac{\sigma}{2\epsilon_o} I was wondering why you can only use a cylindrical gaussian surface and not a cube? Thanks
  41. K

    How Does Gaussian Elimination Help Solve Systems of Equations?

    Homework Statement Using Gaussian Elimination: 3x - 2 y = 5 6x-4y =7The Attempt at a Solution [ 3 -2 5] [6 -4 7] * top row by 1/3 [1 -2/3 5/3] * Top row by -6, add to bottom row: [0 0 -3] So I get this: [1 -2/3 5/3] [0 0 -3] How can 0y equal -3?
  42. S

    Joint density functions (gaussian)

    Would just like a hand with this question If X and Y are independent standard Gaussian random variables (that is, independent N(0, 1) 's ) do the following: (a) Write down the joint probability density function fXX,Y (x, y) of X and Y . I know what the gaussian density function looks...
  43. F

    Basic Gaussian Surface Conceptual Problem

    So I'm pretty sure I have the right answer I just want to make sure I am getting the idea. There is a large solid copper cube. At the center of the cube there is a hollow sphere of radius a. At the center of the hollow sphere there is a charge Q. What is the total charge induced on the...
  44. D

    Spreading of a Free-Particle Gaussian Wave Packet

    Homework Statement It can be shown that the position probability density for the one-dimensional, free-particle Gaussian wave packet can be expressed as shown below, where m is the mass of the particle and L is the position uncertainty at time t=0, vgr = hbar *k0/m, k0 is the average wave...
  45. G

    Can Someone Help with Gaussian Beams?

    Homework Statement 3. A laser scientist tells you that her He-Ne laser can be focused to a spot size of 5.00 microns. When pressed for details, she tells you that she got this number by measuring the Full Width Half Maximum (FWHM) of the intensity profile of the laser waist. a) What is...
  46. D

    Gaussian surfaces can someone help walk me through this problem?

    Gaussian surfaces...need help...can someone help walk me through this problem?? A solid insulating sphere of radius 5 cm carries a net positive charge of 2 μC, uniformly distributed throughout its volume. Concentric with this sphere is a conducting spherical shell with inner radius 10 cm and...
  47. R

    Solve Gaussian Integrals: QFT Explained & Math Book for Physicists

    This is probably an easy question, but my math is not good enough to answer it. For Gaussian integrals: \frac{\int \Pi_i [dx_i] x_k x_l e^{-\frac{x_i A_{ij} x_j}{2}}} {\int \Pi_i [dx_i] e^{-\frac{x_i A_{ij} x_j}{2}}}=A^{-1}_{kl} As far as I understand it, in QFT, Aij is a local operator. So...
  48. X

    Code for Gaussian Elimination

    I have this code for a Gaussian Elimination: As you can see, the input is A and b. I'm trying to figure out how I can make this code work for this Gaussian Elimination: http://img189.imageshack.us/f/50468910.png/ It's 17 rows. The code can do 17 rows. But some of the equations aren't in the...
  49. S

    Gaussian 09 .out file format specification (and freqchk)

    Hello, I was wondering if anyone could tell me where the documentation exists for the output of Gaussian 09 runs. The .out file is plain text and human readable, but I can only decipher parts and units are not always clear. I would have thought the Gaussian website would have an explicit...
  50. X

    Solving Gaussian Elimination Homework

    Homework Statement http://img189.imageshack.us/i/50468910.png/ Homework Equations The Attempt at a Solution I'm wondering how I do this? I don't really know. Supposed to be 17 rows. How many columns and how many each row? No clue! And how do I get the numbers for the matrix here?
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