How to set up gmail SMTP in Wordpress

I'm trying to set up an SMTP gmail server to send emails from my WordPress site. This is what I've got in my wp-config.php:

define( 'SMTP_USER',   '[email protected]' );    // Username to use for SMTP authentication
define( 'SMTP_PASS',   'password' );       // Password to use for SMTP authentication
define( 'SMTP_HOST',   'smtp.gmail.com' );    // The hostname of the mail server
define( 'SMTP_FROM',   '[email protected]' ); // SMTP From email address
define( 'SMTP_NAME',   'My Site Name' );    // SMTP From name
define( 'SMTP_PORT',   '465' );                  // SMTP port number - likely to be 25, 465 or 587
define( 'SMTP_SECURE', 'tls' );                 // Encryption system to use - ssl or tls
define( 'SMTP_AUTH',    true );                 // Use SMTP authentication (true|false)
define( 'SMTP_DEBUG',   1 );                    // for debugging purposes only set to 1 or 2

I put this in my theme's functions.php file:

add_action( 'phpmailer_init', 'send_smtp_email' );
function send_smtp_email( $phpmailer ) {
    $phpmailer-isSMTP();
    $phpmailer-Host       = SMTP_HOST;
    $phpmailer-SMTPAuth   = SMTP_AUTH;
    $phpmailer-Port       = SMTP_PORT;
    $phpmailer-Username   = SMTP_USER;
    $phpmailer-Password   = SMTP_PASS;
    $phpmailer-SMTPSecure = SMTP_SECURE;
    $phpmailer-From       = SMTP_FROM;
    $phpmailer-FromName   = SMTP_NAME;
}

I'm calling wp_mail() in a function like so:

 function invite_others() {
            $team_name = $_GET['team_name'];

            $user_id = get_current_user_id();
            $user = get_userdata($user_id);
            $site = get_site_url();

            $message = "blah blah blah";
            $subject = "blah";
            $admin_email = get_option('admin_email');

            foreach($_POST as $name = $email) {
                if($email != $_POST['invite_others']){ 
                    $headers = "From: ". $admin_email . "\r\n" .
                        "Reply-To: " . $email . "\r\n";
                    $sent = wp_mail($email, $subject, strip_tags($message), $headers);
                }
            }

 }

I get the following error from wp_mail():

SMTP Error: Could not connect to SMTP host

Any help would be appreciated! Thanks